Vector-chromatic p-energy lower-bound conjecture

Let $G$ be a graph, let $p\geq 2$, and let $\chi_v(G)$ denote its vector chromatic number. Let $\mathcal{E}_p^+(G)$ and $\mathcal{E}_p^-(G)$ denote its positive and negative $p$-energies. Vector-chromatic p-energy conjecture. For all $p\geq 2$,

$$\chi_v(G)\geq 1+\max\left\{\frac{\mathcal{E}_p^+(G)}{\mathcal{E}_p^-(G)},\frac{\mathcal{E}_p^-(G)}{\mathcal{E}_p^+(G)}\right\}^{\frac{1}{p-1}}.$$

This would extend the established lower bounds for the vector chromatic number at $p=2$ and $p=\infty$. The supplied text gives counterexamples for the analogous range $0\leq p\leq 1.154$, but does not resolve the conjecture for $p\geq 2$.