The conjectural formula for rho sub 2,3

Let $c1_{2,3}(s,n)$ denote the quantity counting the relevant continuous multiline-queue configurations, let $N$ be the total size parameter used in the paper, and let $f_{(s,s)}$ denote the relevant hook-length count. The conjectural formula.

$$\rho_{2,3}(s,n)=\binom{N-3}{n-1}f_{(s,s)}.$$

The formula is inferred from the displayed numerical table. The source does not provide a proof or resolution.