Eigenvalue interlacing conjecture for the full bivariate Bézier collocation matrix

Let $M_{\mathcal I_d}$ be the full degree-$d$ bivariate Bernstein collocation matrix, and denote its eigenvalues by $\lambda_{i,n}^{\langle d\rangle}$, where the second index records multiplicity. Eigenvalue interlacing conjecture. The spectrum of $M_{\mathcal I_d}$ consists of $d$ distinct real eigenvalues

$$1=\lambda_{1,3}^{\langle d\rangle}>\lambda_{2,3}^{\langle d\rangle}>\lambda_{3,4}^{\langle d\rangle}>\lambda_{4,5}^{\langle d\rangle}>\dots>\lambda_{d,d+1}^{\langle d\rangle}>0,$$

and these eigenvalues satisfy

$$\lambda^{\langle d+1\rangle}_{i,*}\geq\lambda^{\langle d\rangle}_{i,*}\geq\lambda^{\langle d+1\rangle}_{i+1,*},\qquad i=1,2,\dots,d.$$

The claim proposes a Cauchy-type interlacing pattern across consecutive degrees for these collocation matrices. The source presents it as an observed interesting property, and the supplied status is unknown; its conjectural status and notation should be checked.