Minimum determinant conjecture for bivariate Bézier collocation submatrices

Let $d$ be fixed, let $\mathcal I_d$ be the set of weak $3$-compositions of $d$, and for every nonempty subset $\Gamma\subset\mathcal I_d$ let $M_\Gamma$ be the corresponding Bernstein collocation matrix. Let $M_{\mathcal I_d}$ denote the full collocation matrix. Minimum determinant conjecture.

$$\min_{\substack{\Gamma\subset\mathcal I_d\\ \Gamma\neq\emptyset}}\det M_\Gamma=\det M_{\mathcal I_d},$$

where the full determinant is given by the determinant formula in the source. This conjecture strengthens the positivity assertion by proposing the exact smallest determinant among all nonempty principal submatrices. Its status is not resolved by the supplied text.