Karp–Sitnik coefficient-positivity conjecture for rising-factorial Toeplitz determinants

Let $\{f_k\}_{k=0}^{n}$ be a non-negative sequence, let $\alpha,\beta>0$, and define

$$Q^{\alpha,\beta}_n(x):=\sum_{k=0}^{n}f_kf_{n-k}\binom{n}{k}\left[(x+\alpha)_k(x+\beta)_{n-k}-(x+\alpha+\beta)_k(x)_{n-k}\right].$$

Here $(x)_k$ denotes the rising factorial. Karp–Sitnik's coefficient-positivity conjecture. If $f_k^2>f_{k-1}f_{k+1}$ for $k=1,2,\ldots,n-1$ and $n\geq3$, then $Q^{\alpha,\beta}_n(x)$ has positive coefficients at $x^j$ for $j=0,1,\ldots,n-2$.

The conjecture strengthens the known non-negativity result for $Q^{\alpha,\beta}_n(x)$ under log-concavity. Its status is not resolved in the supplied source context.