Sankar's real-rootedness conjecture for valid hook configuration polynomials

For each positive integer $k$, let $h_k(x)$ and $f_k(x)$ be the polynomials associated with $312$-avoiding reduced valid hook configurations, with $h_k(x)=f_k(x-1)$. Sankar's conjecture. For each positive integer $k$, the polynomial $h_k(x)$, and therefore $f_k(x)$, has all real roots. This is posed in the paper's future-work discussion as one of several conjectures, and the supplied material gives no resolution status.