Minimum determinant conjecture for bivariate Bézier matrix submatrices

Let $d$ be a nonnegative integer, let $\mathcal I_d$ be the set of weak $3$-compositions of $d$, and let $N_\Gamma$ be the matrix considered in the paper for a nonempty subset $\Gamma\subset\mathcal I_d$. For $\ell\in\mathbb N$, define

$$n_d:=\begin{cases}\ell^{3\ell},&d=3\ell,\\(\ell+1)^{\ell+1}\ell^{2\ell},&d=3\ell+1,\\(\ell+1)^{2\ell+2}\ell^\ell,&d=3\ell+2.\end{cases}$$

Exact minimum conjecture.

$$\min_{\substack{\Gamma\subset\mathcal I_d\Gamma\neq\emptyset}}\det N_\Gamma=n_d.$$

This gives a proposed closed form for the smallest determinant among the matrices $N_\Gamma$, refining the determinant-minimization problem. The supplied text does not establish its status.