The exact-variable Finiteness Conjecture for nonsingular unsatisfiable hitting clause-sets

Let $[?]$ be the supremum of the number $n(F)$ of variables among nonsingular unsatisfiable hitting clause-sets $F$ of deficiency $k$:

$$\maxnhitdef(k)=\sup\{n(F):F\in\Uclashnsi{\delta=k}\}.$$

Exact-variable Finiteness Conjecture. For every $k\in\mathbb{N}$,

$$\maxnhitdef(k)<+\infty.$$

This is the formal version of the Finiteness Conjecture for the nonsingular subclass. The paper’s abstract states that the case $k=3$ is proved, with the cases $k\leq 2$ already known; the general assertion remains open.