Let $r$ be a positive integer and let $${\left\langle\begin{array}{l}n \\k\end{array}\right\rangle}_r=\prod_{1 \le i \le k, 1 \le j \le r} \frac{n - i + j}{(k - i) + (r - j) + 1} = \prod_{1 \le i \le k} \frac{(n - i + 1)^{\overline{r}}}{(k - i + 1)^{\overline{r}}}, $$ where $x^{\overline{r}}$ denotes the rising factorial $ x(x + 1) \dots (x + (r-1)),$
be the elements of the $r-$Hoggatt triangle which generalizes Pascal’s triangle.
Consider the Hankel determinants $$d_k(r,m,n)= \det{ \left({\left\langle\begin{array}{l}k+i+j\\ {m}\end{array}\right\rangle_r } \right)}_{i,j = 0}^{n - 1}.$$ Computations suggest that $$\sum_{k\ge 0}d_{k+m-n+1}(r,m,n)x^k =(-1)^{\binom{n}{2}} \frac {A_{r,m,n}(x)}{(1-x)^{n(rm-n+1)+1},}$$ where $ A_{r,m,n}(x)$ for $m \ge {n-1}$ is a palindromic and unimodal polynomial of degree $(nr-1)m-r-n^2+n+1$ with positive coefficients which is also gamma positive.
Any idea how to prove these properties of $ A_{r,m,n}(x)$?