Entropy lower-bound conjecture for positive-genus self-shrinkers

Let $\Sigma\subset \mathbb R^3$ be a closed self-shrinker of positive genus, and let $\Sigma_C\subset \mathbb S^3$ be the Clifford torus. One should have

$$1.57 \approx \frac{\pi}{2}=\bar{\lambda}_{LY}^2[\Sigma_C]\leq \lambda_{CM}^2[\Sigma].$$

This conjecture is motivated by the Willmore-conjecture result for positive-genus surfaces in $\mathbb S^3$ and the comparison between stable conformal volume and Colding-Minicozzi entropy. The source does not report a resolution.