The asymptotic Christoffel-like function conjecture for tensor-product polynomial spaces

Let $\boldsymbol{\Omega}$ be the compact support of a positive Borel probability measure $\mu$ on $\mathbb{R}^d$, and let $f$ be its positive, continuous density with respect to Lebesgue measure on $\boldsymbol{\Omega}$. Write $s(n,d)_\infty$ for the normalization used for the tensor-product polynomial space and $\Lambda_{\otimes n}^{\mu}$ for the associated Christoffel polynomial. Then, for $\mathbf{z} \in \mathbb{R}^d$, the conjectured asymptotic formula.

$$\lim_{n \to \infty} \frac{1}{s(n,d)_\infty} \Lambda_{\otimes n}^{\mu}(\mathbf{z}) = \frac{\omega_{E, \infty}(\mathbf{z})}{f(\mathbf{z})},$$

where $\omega_{E, \infty}: \mathbb{R}^d \to \mathbb{R}_{\geq 0}$ depends on $\mathbf{\Omega}$ but not on $f$. The conjecture seeks a tensor-product analogue of the known asymptotic behavior for total-degree Christoffel polynomials; establishing the limit and identifying its support-dependent factor remain open in the stated generality.