Sankar's three-dimensional Catalan formula for reduced valid hook configurations

Let $\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))$ denote the set of $312$-avoiding reduced valid hook configurations with $k$ hooks on permutations with $2k+i$ points. Define

$$f_k(x)=\sum_{i=0}^{k-1}\left|\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))\right|x^i,$$

and let $h_k(x)=f_k(x-1)$. Sankar's conjecture. For all $k\geq 1$,

$$f_k(0)=h_k(1)=\left|\operatorname{RedVHC}_k(\operatorname{Av}_{3k}(312))\right|=2\frac{(3k)!}{k!(k+1)!(k+2)!}.$$

The right-hand side is the $k$th three-dimensional Catalan number; the source presents this as one of Sankar's conjectures about these counting polynomials, and the supplied material gives no resolution status.