The asymptotic scarcity conjecture for collapsible hypertrees

Let ${\mathcal T_{n,d}}$ denote the family of $n$-vertex $d$-hypertrees, and let ${\mathcal C_{n,d}}$ denote the set of $n$-vertex $d$-collapsible $d$-hypertrees. Here, a $d$-collapsible $d$-hypertree is a $d$-hypertree that can be reduced by elementary $d$-collapses, each removing an exposed $(d-1)$-face and its unique containing $d$-face. Asymptotic scarcity conjecture. For every $d\ge 2$, asymptotically almost none of the $d$-hypertrees are $d$-collapsible; namely,

$$\frac{|{\mathcal C_{n,d}}|}{|{\mathcal T_{n,d}}|}\to 0$$

as $n\to\infty$. This conjecture asserts that, despite collapsibility characterizing ordinary trees, $d$-collapsible hypertrees form a vanishing fraction of all $d$-hypertrees in every higher dimension.