Yes, this sort of too-near objects produce retinal rivalry. My
recipe is masking them out. STMANI3 has feature for this as shows
this example from the STMANI3 user's manual where an unwanted rock
appears in the right image:
In fact, there is one case where you would need it, even with
a perfect hardware: with an hyper-stereo near first plan. When
shooting from too near positions for the first plan regarding
the space between lens (too large for that distance), parts of
the image will be nearer on one side than the other side, and
when say nearer say larger. You will then be obliged to
balance the alignment to keep with these larger/smaller parts,
as we did with the Spider-Man example.
I was speaking about hyper-stereo objects in near foreground.
But, this rock is not a stereo at all.