Sankar's positivity conjecture for the transformed valid hook configuration polynomial

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. The polynomial $h_k(x)$ has strictly positive coefficients. The statement concerns the coefficient sign pattern of the transformed counting polynomial, and the supplied material gives no resolution status.