Sankar's weighted tennis ball formula for the linear coefficient

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 linear coefficient of $h_k(x)$ is given by the $(k-1)$st weighted tennis ball number. The supplied material does not define the weighted tennis ball numbers or give a resolution status for this conjecture.