Unimodality conjecture for the coefficients $b(n_k,\ldots,n_1)$

Let $k\geq 2$ be an integer and let $n_k,\ldots,n_1$ be non-negative integers. Let $b(n_k,\ldots,n_1)$ denote the polynomial constructed in Theorem 3. Unimodality conjecture. For every such choice of $k$ and $n_k,\ldots,n_1$, the polynomial $b(n_k,\ldots,n_1)$ is unimodal. The conjecture is motivated by empirical evidence; the source gives no proof or resolution, so the unimodality of these polynomials remains open.