Sure Bastiaan! You write that you have a "gated" stroke but then surmise that your problem "starts" in the downswing. That's not correct: your problem starts in having a "gated" stroke AND in the downswing.
Start the putter back with the LEAD arm, not the rear hand. It might be hard to tell the difference, so start with the lead side having a little more muscle tone and grip pressure than the rear hand, and then MOVE the putter back with the lead arm-and-hand as a unit. This REDUCES the "gating" -- why would you want to gate the putter face open in the backstroke -- all that does is create a problem for the thru-stroke, and the thru-stroke is the ONLY stroke that matters?
So starting back with the lead side reduces the inside gating problem.
What also reduces it is a flat thumb print of the lead hand pressed flat to the flat surface of the putter handle. The flatness of the handle is the same left-right direction as the putter face (assuming the grip was properly put on). So KEEPING the thumb print from TWISTING in the back and thru stroking also manages the putter face angle and prevents putter face opening and closing.
In the THRU-SWING (not really the "downswing", which is only part of the thru-swing), the PURPOSE is to move the putter face with the same SQUARE orientation aimed at address thru impact on the ball, and to move the putter face STRAIGHT on the line aimed at address -- this then is what rolls the balls the same line the putter face aims at address.
It does not require SWEETSPOT impact on the back of the ball for the above two features -- ANY piece of the putter face that meets the ball will do, toe or heel or sweetspot or anywhere else, so long as when that piece of the putter meets the ball it is moving STRAIGHT thru the ball on the aim line and the putter face is then SQUARE and STAYS SQUARE thru impact (in physics, about 1/10th of an inch, but in motor movement about 2-4 inches).
The THRU-SWING to accomplish this rolling the ball exactly wherever the putter face was aiming at address might start from a top-of-the-backstroke position that is slightly inside the aim line or exactly on the aim line (and rarely and poorly even outside the aim line). If the putter face aim is managed properly, none of that matters, since the THRU-SWING will simply be STRAIGHT ACROSS IN FRONT OF THE BODY, with hands sliding straight on a path that is parallel to the aim line at address. Then SOME PART of the putter face impacts the ball 1) moving straight down the aim line, 2) with the same square face the putter had at address.
If you TWIST the putter face OPEN in the backstroke, why did you do that? If you TWIST the putter face open, either you twisted the thumb print, or rolled the forearm, or rotated the chest open. So don't rotate the chest in the backstroke, don't roll the forearms in the backstroke, and don't twist the thumb print in the backstroke.
This really isn't hard, as all it requires is using the pectoral muscle of the lead side to draw the lead arm and hand and putter back into the backstroke, while not using OTHER muscles to roll the forearm or rotate the chest, and while also just sliding the thumb print back and thru without twisting.
Practice the THRU-STROKE along the base of a wall, focusing upon the toe of the putter staying the same distance back from the wall in the THRU-STROKE from just before impact and then all the rest of the way thru to at least one foot past impact, and focusing upon the putter face staying SQUARE and hence aimed parallel to the wall and the heel-toe line staying perpendicular to the wall.
EVEN IF the backstroke gates a little to the inside, but without the thumb print twisting, the THRU-STROKE moves the hands STRAIGHT parallel to the aim line. In the preferred setup, the chest and shoulders parallel the aim line of the putter face at address, so the hands then swing straight along in front of the chest and shoulders, parallel to their sideways orientations. This means that the hands REMAIN the same distance off the thighs throughout the stroke that they have at address. The hands do not drift closer to the body in the thru swing or drift further from the body in the thru swing than the distance they start out.
IF the hands drift closer or farther from the body during the stroke, either your elbows are changing shape or your armpits are changing shape. Drifting in closer to the body is a PULL and that happens by using the rear hand, so swing the THRU-STROKE with the lead arm and hand. If you swing the stroke with the lead arm and hand, the danger remains of the stroke headed out away from parallel into something like a PUSH. That's the benefit of the practice of strokes along a wall -- you cannot PUSH without hitting the wall.
If you hit the wall, that's because either your elbows flex in the thru-stroke or your armpits move upward forward outward as your elbows move forward from your chest. If the angle that your upper arm hangs off the chest stays the same during the THRU-STROKE, the hands will move straight across the body parallel to the shoulders. So, hang the arms and set some nice muscle tone and use dead hands to swing, as this reduces or eliminates elbow changes in the thru-stroke, and also maintain the same angle at the armpits for how the upper arms hang off the shoulders, as this controls the straightness of the hands moving in the THRU-STROKE straight parallel to the shoulders. With the thumb print not twisting open or closed but merely sliding straight with the hands motion, your problem gets a lot smaller.
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