The asymptotic vanishing-ratio conjecture for higher diversities

For each $k$, let $m_{\ell}(k)$ denote the extremal quantity corresponding to the $\ell$-diversity introduced in the paper. Asymptotic vanishing-ratio conjecture. For all $\ell\geq 0$,

$$\lim_{k\rightarrow\infty}\frac{m_{\ell+1}(k)}{m_{\ell}(k)}=0.$$

This conjecture predicts that successive diversity levels become asymptotically negligible relative to the preceding level as the uniformity $k$ tends to infinity; the source presents it as an open problem.