Generalized Laguerre inequality for Speyer's generating polynomial

Let $h_n(x;t)$ denote the generating polynomial associated with Speyer's $g$-polynomials, viewed as a polynomial in $x$, and let $f^{(k)}(x)$ denote the $k$-th derivative of $f(x)$. For $r\geq 1$, define the generalized Laguerre expression

$$L_r(f(x))=\frac{1}{2}\sum_{k=0}^{2r}(-1)^{r+k}\binom{2r}{k}f^{(k)}(x)f^{(2r-k)}(x).$$

Generalized Laguerre conjecture. Given $t>0$, the polynomial $h_n(x;t)$ satisfies

$$L_r(h_n(x;t))\geq 0$$

for $r\geq 1$. Generalized Laguerre inequalities characterize important aspects of the Laguerre–Pólya class; the source proposes this assertion after establishing the classical Laguerre inequality, and its status is otherwise unresolved here.