Alon et al.'s VC-dimension conjecture for completions of gap Hamming distance

Let $\operatorname{GHD}_k^n$ be the partial sign matrix for the gap Hamming distance problem, and let $A$ be any total sign matrix obtained by replacing every $*$ entry of $\operatorname{GHD}_k^n$ arbitrarily by $-1$ or $1$. Let $\operatorname{VC}(A)$ denote the VC dimension of $A$. Alon et al.'s conjecture. There exists a function $g(k)\to\infty$ such that, for every such completion $A$,

$$\operatorname{VC}(A)\geq g(k).$$

A positive answer would show that learnability is a topological phenomenon for total concept classes and would imply a strong form of the open problem concerning learnability of disambiguations of half-spaces with margin. The source presents this as a conjecture attributed to Alon et al.; no resolution is given.