....Not sure what to search for, tried tire size, difference, limited slip and blowout.

So, the back story is I had a tire blowout on my SCREW (10 ply Michelin) and instead of trashing the other 3 all with about 25K on them I found a used exact replacement.

Here is the question.....the replacement is 12/32 and the other 3 (mine) are 9/32 to 10/32. I have a limited slip diff and am worried that I will do damage to it by running this tire on the rear. I also don't want to run on the front even though it is not limited slip due to possible wear on my newer front end parts and alignment. So is there an allowable difference in tires side to side for LSD's and if so what is it. The local tire shops all give me different answers ranging from NO difference allowed to 3-4/32's allowed. HELP