Linear singular-index conjecture for unsatisfiable hitting clause-sets

Let $\mathcal{U}\mathcal{HIT}_{\!\delta=k}$ be the class of unsatisfiable hitting clause-sets of deficiency $k$. Let $\operatorname{singind}(F)$ denote the singular index of $F$, and let $\operatorname{varsing}(F)$ be the set of singular variables of $F$. Linear singular-index conjecture. For every $k \in \mathbb{N}$ there are $a \in \mathbb{N}$ and $\alpha \in \mathbb{R}_{>0}$ such that, for all $F \in \mathcal{U}\mathcal{HIT}_{\!\delta=k}$ with $\operatorname{singind}(F) \geq a$,

$$\lvert\operatorname{varsing}(F)\rvert \geq \alpha \cdot \operatorname{singind}(F).$$

The conjecture asks for a quantitative relation between the singular index and the number of singular variables. The source presents it as an open question in its discussion of singular tuples; no resolution is supplied.