The conjectural formula for the one-wrapping count gamma

Let $b3^\ell(s,n)$ denote the number of continuous multiline queues with the specified one-wrapping configuration, and let $f_{(n-1,s+1)}$ denote the relevant hook-length count. The conjectural formula.

$$\gamma^\ell(s,n)= (\ell-1)\binom{n+2s-\ell}{s-\ell}f_{(n-1,s+1)}.$$

This formula is proposed after numerical checks because no closed form for the preceding sum was found; the source gives no proof or resolution.