Constant-count conjecture for high-rank string C-group representations of symmetric groups

Let $S_n$ be the symmetric group, and let a string C-group representation mean a string C-group representation of a specified rank. For an integer $i$ satisfying

$$1\leq i\leq (n-3)/2,$$

consider representations of rank $n-i$ of $S_n$.

Constant-count conjecture. The number of string C-group representations of rank $n-i$ for $S_n$ with $1\leq i\leq (n-3)/2$ is a constant independent of $n$.

This conjecture concerns the enumeration of representations in the high-rank range and is motivated by computational data for symmetric groups. The surrounding enumeration problem is solved for the four highest values of the rank and sufficiently large $n$, but the stated uniform constancy remains open.