Optimal cell-average decomposition equality for tensor-product and total-degree polynomial spaces

Let $\theta\in[-1,1]$ and let $\mathbb V^k$ denote either the tensor-product polynomial space $\mathbb Q^k$ or the total-degree polynomial space $\mathbb P^k$, with $k\in\mathbb N_+$. Let $\overline{\omega}_\star(\theta,\mathbb V^k)$ and $\phi^\star(\theta,\mathbb V^k_+)$ be the quantities defined for the corresponding positive-polynomial optimization problem. A critical positive polynomial is a nonzero polynomial attaining the critical value. Optimal cell-average decomposition conjecture. For either choice of $\mathbb V^k$,

$$\overline{\omega}_\star(\theta,\mathbb V^k)=\phi^\star(\theta,\mathbb V^k_+)\qquad\forall\theta\in[-1,1],$$

and there always exists a critical positive polynomial for $\phi^\star(\theta,\mathbb V^k_+)$. The claim is presented as a conjecture based on extensive numerical experiments and is intended to guide the understanding, construction, and verification of two-dimensional optimal cell-average decompositions.