Squared critical positive polynomial conjecture for optimal cell-average decomposition

Let $\theta\in[-1,1]$, let $\mathbb V^k$ be either $\mathbb Q^k$ or $\mathbb P^k$, and let $\mathbb V^k_+$ denote the corresponding cone of positive polynomials. For $k\in\mathbb N_+$, write $\lfloor k/2\rfloor$ for the integer part of $k/2$, and let $\phi(q^2;\theta)$ be the objective associated with the positive-polynomial optimization problem. A critical positive polynomial is a nonzero polynomial attaining the critical value. Squared critical polynomial conjecture. For either choice of $\mathbb V^k$,

$$\phi^\star(\theta,\mathbb V^k_+)=\phi^\star\left(\theta,(\mathbb V^{\lfloor k/2\rfloor})^2\right):=\inf_{q\in\mathbb V^{\lfloor k/2\rfloor}}\phi(q^2;\theta)\qquad\forall\theta\in[-1,1],$$

and there exists a critical positive polynomial $p^\star=q_\star^2$ with $q_\star\in\mathbb V^{\lfloor k/2\rfloor}\setminus\{0\}$. This is reported as a further numerical finding for constructing and verifying two-dimensional optimal cell-average decompositions; the excerpt gives no proof or resolution.