Mitra–Pyatov conjecture for the number of HSFPL diagrams

Let an FPL diagram of size $L$ have size $(L-1)\times L/2$ when $L$ is even and size $L\times (L-1)/2$ when $L$ is odd. Let $d$ be its depth, and let $S(L,d)$ denote the total number of HSFPL diagrams of size $L$ and depth $d$. Here $(a)_k$ denotes the rising factorial and $\Gamma$ denotes the gamma function. Mitra–Pyatov conjecture. The total number is

$$S(L,d)=\prod_{k=0}^{d}\frac{\Gamma(L-k+1)}{2^k(1/2)_k\Gamma(L-2k+1)}\frac{\Gamma\left(\frac{2L+2k+3}{6}\right)\Gamma\left(\frac{L-2k+3}{3}\right)}{\Gamma\left(\frac{2L-k+3}{6}\right)\Gamma\left(\frac{2L-k+6}{6}\right)}.$$

Assuming the RS conjecture, this formula was subsequently proved, so the conjecture is resolved.