Asymptotic growth conjecture for the number of distinct multinomial coefficients

Let $M(n)$ denote the number of distinct multinomial coefficients arising from partitions of $n$. Asymptotic-growth conjecture. There exists a positive constant $C$ such that

$$\lim_{n\to\infty}\frac{\log M(n)}{\sqrt n}=C.$$

This conjecture concerns the order of magnitude of $M(n)$. The paper gives conditional bounds on $C$ from the preceding partition lower-bound conjecture, but does not establish the asserted limit.