Pyatov's conjecture for the HSFPL nest generating function

For half-turn symmetric FPL diagrams with $L=2n$ external terminals, let $P(L,m)$ count diagrams with $m+1$ nests and define

$$\mathcal{P}(L;z)=\sum_{m=0}^{n-1}P(L,m)z^m.$$

Let $Z_{\rm HTSFPL}(2n)$ be the total number of half-turn symmetric FPL diagrams of size $2n$. Pyatov's conjecture. The nest generating function is

$$\mathcal{P}(2n;z)=Z_{\rm HTSFPL}(2n)\frac{3n}{4n^2-1}\,{}_3F_2\left(\begin{array}{c}\frac{3}{2},1-n,1+n\\2-2n,2+2n\end{array};4z\right).$$

The paper attributes this conjecture to de Gier; it is stated only for $L=2n$ and $d^*=0$, and the supplied text gives no resolution, so it remains open.