Let $S_n$ denote the set of partions of $n$ such that every part is greater than 1. Partitions $(x_1,\ldots,x_k), (y_1,\ldots,y_l) \in S_n$ are said to have almost equal product if $$\prod_{i=1}^k (x_i+1) = \prod_{i=1}^l (y_i+1)$$.

For example if $n = 14$ the partitions (3,3,8) and (2,5,7) are almost equal since (3+1)(3+1)(8+1) = (2+1)(5+1)(7+1).

Now if we denote by $S'_n$ the largest subset of $S_n$ not containing almost equal partitons,then I would like to find the asymptotic value of $|S'_n|$. I believe $|S'_n|$ is at least subexponential in $n$ but do not know how to prove this. Is there any way to perhaps find a bound on the number of pairs of almost equal partitions and take it from there?