Sprites moving (shaking) during animation while they should not

I’m currently working on a 2D project where we have sprites which are never moving, and we have a fixed camera. However, when animated, the sprites are moving in a random direction from 0.5 or 1 pixel around the place they should be. We have tried many things on the sprite but it looks like the problem comes from unity, as it doesn’t appear if we try to make a gif from the animation for example.

I put the filter mode to bilinear after the problem showed up and it practically solves the problem for the eye but it is still moving.

Here you can see a video of the problem

Any idea of what is missing ?

@infenix, see this thread: https://forum.unity.com/threads/setting-for-pixel-game-sprite-seem-to-move-when-movement.501505/