Exact asymptotic constant for the number of distinct multinomial coefficients

Let $M(n)$ denote the number of distinct multinomial coefficients arising from partitions of $n$. Exact-asymptotic-constant conjecture.

$$\lim_{n\to\infty}\frac{\log M(n)}{\sqrt n}=\frac{\pi}{3}\sqrt{2}.$$

The paper explains that this would follow from the two proposed partition bounds, which would replace the earlier conjecture asserting only the existence of a positive limiting constant. It remains open in the source.