The maximal-girth dimension conjecture for l-CM simplicial complexes

Let $\Delta$ be an $l$-CM simplicial complex of maximal girth. Let $c$ denote the parameter used for its dimension in the paper, and suppose that $\Delta$ is not the $r$-skeleton of the $(l+r-1)$-simplex for any $r$. The maximal-girth dimension conjecture. Then

$$c \geq l-1.$$

The conjecture asserts a lower bound on the dimension-related parameter $c$ for maximal-girth $l$-CM complexes, apart from the specified skeletons. The source gives no resolution, and the displayed assertion is also repeated in a weaker large-vertex form.