The answer to the question in the title is “YES!”. Sniper’s Hide has recently released a video, where they experimentally prove that and explain why it happens. The host of the channel shows that shooting from the bench (sitting) results in 20 fps decrease of average muzzle velocity compared to shooting from the prone position. When shooting from the bench he gets an average muzzle velocity of 2696 fps and from the prone position, MV is 2717 fps. Here is the video with that experiment:

So actually, it is not the shooter’s position or stance itself that affects the muzzle velocity, but the ability to manage the recoil from any particular position. The better you control (resist) the recoil, the more is the muzzle velocity. My understanding is that less energy spent on moving the recoiling gun means more energy is spent to propel the projectile. So according to this logic, the highest muzzle velocity will be achieved if the gun is firmly fixed and doesn’t move at all during shooting. And the opposite should be correct too: the more unstable is the position and the more the gun moves upon recoil, the more loss in MV you’ll have.

Sniper’s Hide also proves that it is about recoil management by shooting from the bench again, but this time having both elbows square on the table and “leaning” on the gun. This time they get results almost identical to that of the prone position.

For most of the shooters, this 20 fps difference won’t even matter. But when it comes to long range precision shooting, it does matter.