Myers' asymptotic monotone-subsequence conjecture

Let $k$ be a positive integer. For a permutation $\tau$ of $[n]=\{1,\dots,n\}$, let $m_k(\tau)$ denote the number of monotone subsequences of length $k+1$. Myers' asymptotic conjecture. As $n\to\infty$, every permutation of $[n]$ has at least

$$(1+o(1))\frac{1}{k^k}\binom{n}{k+1}$$

monotone subsequences of length $k+1$. This is a weaker asymptotic form of Myers' exact minimum conjecture, and it remains open in the generality stated.