Sublinear multiplicity bound for the one-half eigenvalue of the drifted Jacobi operator

Let $\Sigma^2\subset \mathbb{R}^N$ be a two-dimensional self-shrinker, let $\mathcal{L}$ denote its drifted Jacobi operator, and let $\lambda(\Sigma)$ be its entropy. Fix $\gamma$ as in the surrounding hypotheses. The multiplicity of the eigenvalue $\frac{1}{2}$ of $\mathcal{L}$ is conjectured to satisfy

Sublinear multiplicity conjecture. There exist $\alpha<1$ and a constant $C_{\alpha}=C_{\alpha}(\alpha,\gamma)$ such that

$$\operatorname{mult}_{\mathcal{L}}\left(\frac{1}{2}\right)\leq C_{\alpha}\,\lambda^{\alpha}(\Sigma).$$

Such a bound would, via the cited result, yield the paper's entropy theorem without the assumption $N\geq C\,\lambda(\Sigma)$.