Legendrian Willmore conjecture for positive-genus surfaces

Let $\mathbb{S}^5\subset\mathbb{C}^3$ be the unit sphere, and let $\Sigma_H\subset\mathbb{S}^5$ be the hexagonal Legendrian minimal torus. For a Legendrian surface $\Sigma\subset\mathbb{S}^5$, write $\mathcal{W}_{CR}[\Sigma]$ for its CR-Willmore energy. Legendrian Willmore conjecture. If $\Sigma\subset\mathbb{S}^5$ is a Legendrian surface of positive genus, then

$$\mathcal{W}_{CR}[\Sigma]\geq\mathcal{W}_{CR}[\Sigma_H]=\frac{4\sqrt{3}}{3}\pi^2.$$

Equality should hold if and only if $\Sigma$ differs from $\Sigma_H$ by an isometry of $\mathbb{S}^5$. This is the Legendrian analogue of the Willmore conjecture; for tori it is equivalent to a conjecture of Wang, and it is related to questions about Willmore-type energies of Lagrangian tori in $\mathbb{CP}^2$.