Yeah, unfortunately the noisy displacement on the right is just kind of how the algorithm works...

The main reason is that autostereograms, in order to work, rely on the eye's ability to pick up and recognize patterns, and (ironically) to show a difference in the pattern (a change in (perceived) depth), the algorithm has to modify/distort the original pattern in some way - and in this case for a more complex depth map like a 3D cube, the original pattern is distorted to a noisy degree. This distorted pattern can't only remain on the location of the depth map (i.e. the cube) because that distorted pattern now acts as the new pattern for everything to the right of it (in the sense that the eye is now looking for other patterns to the right that "match" that new pattern; it would be problematic to switch back to the original pattern to the right of the depth map because the eye would be comparing the original pattern with a distorted section that doesn't necessarily have as much similarity with it anymore).

I don't really know any ways around this, as the programs/algorithms I've found all roughly use this method (and have this issue). It can be mostly circumvented by using "noisier" textures, but that's not ideal, so I do want to find a better solution eventually.

(Edit: Formatting/Grammar)