The variance bound for higher-degree defining equations

Let $X \subset \mathbb{P}$ be a smooth well formed Fano weighted complete intersection of dimension at least $2$ which is not an intersection with a linear cone. Let $s_2(X)$ denote the number of defining degrees greater than $2$, and let $\operatorname{var}(X)$ denote the variance of $X$. Variance bound conjecture. Then

$$s_2(X) \leq \operatorname{var}(X).$$

This conjectural estimate would provide the generalization of the paper's finiteness result for series of smooth Fano weighted complete intersections. Its resolution status is not specified in the supplied text.