Lower-bound and minimizer conjecture for Lie algebra representation potentials

Let $\mathfrak{a}$ be a simple Lie algebra, let ${\mathbf d}\geq 1$, and let $\psi_{\mathbf d}$ be the real-valued polynomial obtained by restricting the potential $\Psi$ to the space of $\star$-representations. Write $\operatorname{Rep}^\star_{\operatorname{min}}(\psi_{\mathbf d})$ for its set of absolute minima, and let $\operatorname{\mathsf{Irrep}}^\star_{\mathbf d}{\mathcal U}\mathfrak{a}$ denote the set of irreducible Lie algebra representations $\mathfrak{a}\to\operatorname{Mat}^{\operatorname{skew}}_{\mathbf d}$. The lower-bound and minimizer conjecture. For every simple Lie algebra $\mathfrak{a}$ and every ${\mathbf d}\geq 1$, the function $\psi_{\mathbf d}$ is bounded from below. If $\operatorname{\mathsf{Irrep}}^\star_{\mathbf d}{\mathcal U}\mathfrak{a}$ is nonempty, then

$$\operatorname{Rep}^\star_{\operatorname{min}}(\psi_{\mathbf d})=\operatorname{pr}_{\mathcal U}^*(\operatorname{\mathsf{Irrep}}^\star_{\mathbf d}{\mathcal U}\mathfrak{a}).$$

This predicts that the absolute minima are exactly the irreducible unitary representations. The source provides no resolution evidence, so the claim is recorded as open.