Di Francesco's determinant evaluation for twenty-vertex configurations

Let $n$ be a positive integer. For integers $\alpha$ and $p$, define

$$\binom{\alpha}{p}=\begin{cases}\dfrac{\alpha(\alpha-1)\cdots(\alpha-p+1)}{p!},&p\geq0,\\0,&p<0.\end{cases}$$

Di Francesco's conjecture.

$$\det_{0\leq i,j\leq n-1}\left(2^i\binom{i+2j+1}{2j+1}-\binom{i-1}{2j+1}\right)=2\prod_{i=1}^n\frac{2^{i-1}(4i-2)!}{(n+2i-1)!}.$$

This determinant counts the relevant twenty-vertex configurations and corresponding domino tilings of Aztec triangles. The source paper proves the conjecture, although the supplied parser does not provide a resolution status; the claim is therefore recorded as open pending verification of that proof.