Lower bound for the number of distinct multinomial coefficients

Let $M(n)$ be the number of distinct multinomial coefficients arising from partitions of $n$, and let $p^*(n)$ be the total number of partitions of $n$ into parts that are either at most $6$ or multiples of $3$ (or both). Lower-bound conjecture. For every integer $n\geq 0$,

$$M(n)\geq p^*(n).$$

This conjecture gives a combinatorial lower bound for the number of distinct multinomial coefficients and is part of the paper's proposed description of the growth of $M(n)$.