Index lower-bound conjecture for branched CMC immersions

Let $u:\Sigma^2\to N^3$ be a branched CMC $h$-immersion, where $N^3$ has strictly positive sectional curvature, and let $g$ denote the genus of $\Sigma$. Write $i_{\mathcal{A}_h}$ and $n_{\mathcal{A}_h}$ for the index and nullity of the relevant constrained second-variation operator. Index lower-bound conjecture. There exists a constant $C=C(N)<\infty$ such that

$$C\left((1+h^2)\operatorname{Area}(\Sigma)+g\right)\leq i_{\mathcal{A}_h}+n_{\mathcal{A}_h}.$$

For $N=\mathbb{R}^3$ or $N=T^3$ a flat three-torus, the analogous expected bound is

$$C\left(h^2\operatorname{Area}(\Sigma)+g\right)\leq i_{\mathcal{A}_h}+n_{\mathcal{A}_h}.$$

The conjecture asserts that the index plus nullity is bounded below linearly by the Willmore-type energy and genus, complementing known index lower bounds in terms of genus. The stated flat-ambient version is presented as an expectation rather than an asserted theorem, and the source gives no resolution.