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. Define

$$\mu=\mathbb{E}[L^1_{m,n}(\pi)].$$

For each $r\ge 1$, define

$$c_r=\begin{cases}\dfrac{r!}{2^{r/2}(r/2)!}& r\text{ even},\\[4pt]\dfrac{r!}{3\cdot 2^{(r-1)/2}((r-3)/2)!}& r\text{ odd}. \end{cases}$$

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

$$\mathbb{E}[(L^1_{m,n}(\pi)-\mu)^r]=c_r m^{\left\lfloor r/2\right\rfloor}+O\left(m^{\left\lfloor r/2\right\rfloor-1}\right).$$

The conjecture is based on computational evidence for higher moments. Together with the known asymptotic standard deviation $\sigma\sim m^{1/2}$, it would imply convergence of the standardized moments to the Gaussian moments, and hence convergence in distribution to a standard normal law.