Odd-part dominance for elementary symmetric products of partition parts

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_i(\lambda)$ be the partition whose parts are all products $\lambda_{i_1}\cdots\lambda_{i_i}$ with $1\leq i_1<\cdots<i_i\leq\ell$, and set $\operatorname{pre}_i(\lambda)=\emptyset$ when $\ell<i$. Let $o_i(n)$ be the number of partitions in $\operatorname{pre}_i(\mathcal P(n))$ with odd parts, and let $d_i(n)$ be the number with distinct parts. Odd-part dominance conjecture.

$$o_2(n)\geq d_2(n)\quad(n\geq0),\qquad o_3(n)\geq d_3(n)\quad(n\geq21).$$

These inequalities are based on numerical evidence and relate the parity and distinctness of partitions formed from elementary symmetric products; the source does not establish them.