Family D conjecture for $D_{-1,2r}(2n)$

Let $\mu$ be an indeterminate, let $n\geqslant1$ and $r\geqslant0$ be integers, and define the rising factorial $(a)_k=a(a+1)\cdots(a+k-1)$. Let $D_{-1,2r}(2n)$ be the determinant associated with Family D. Family D conjecture.

$$D_{-1,2r}(2n)=\prod_{i=0}^{n-1}R_{-1,2r}(i),$$

where $R_{-1,2r}(n)$ is the piecewise expression given in the source for $n>r$, $n=r$, and $n<r$. Equivalently, $R_{-1,2r}(n)=D_{-1,2r}(2n+2)/D_{-1,2r}(2n)$. Families C and D are the cases for which the authors state that they have not found a proof, because their tiling regions do not reduce to a single triangular hole.