Non-central moment conjecture for continuously increasing subsequences

Let $\mathfrak{S}_{m,n}$ be the set of multiset permutations under consideration, let $\pi$ be uniformly random in $\mathfrak{S}_{m,n}$, and let $L^1_{m,n}(\pi)$ be the continuously increasing subsequence statistic.

Non-central moment conjecture. For all $r\ge 1$, if $n$ is sufficiently large in terms of $m$, then

$$\mathbb{E}[L^1_{m,n}(\pi)^r]=m^r+{r+1\choose 2}m^{r-1}+O(m^{r-2}).$$

The source presents this as a possible first step toward proving the central moment conjecture, since a better understanding of the non-central moments could help establish the asserted asymptotics for centered moments.