Elizalde–Noy maximality conjecture for consecutive pattern avoidance

Let $S_k$ denote the set of permutations of length $k$, and let $\alpha_p(n)$ be the number of permutations of length $n$ avoiding the consecutive pattern $p$. For a fixed pattern length $k$, write

$$\sigma=1 2 \ldots k.$$

Elizalde–Noy maximality conjecture. The increasing pattern $\sigma$ is the maximal pattern, in the sense that

$$\alpha_{\sigma}(n) \geq \alpha_p(n)$$

for all $p\in S_k$ and all $n$. This conjecture concerns the extremal enumeration of consecutive pattern avoidance; the source attributes it to Elizalde and Noy and gives no resolution.