Distinct-part bound conjecture for the missing-integer parity difference

Let $P(n,m)$ denote the number of partitions of $n$ with $m$ missing integers, and define

$$\mathcal{M}_e(n)=\sum_{k=0}^{\lfloor \frac{n-1}{2} \rfloor}P(n,2k),\qquad \mathcal{M}_o(n)=\sum_{k=1}^{\lfloor \frac{n}{2} \rfloor}P(n,2k-1).$$

Let $q(n)$ denote the number of partitions of $n$ into distinct parts. Distinct-part bound conjecture. For $n>10$,

$$\mathcal{M}_e(n)-\mathcal{M}_o(n)<q(n).$$

The conjecture gives an upper bound for the parity difference in terms of distinct-part partitions. The source offers no proof or resolution.