Inequalities between parity-separated partition functions

Let $\overline{\mathcal{OE}}(n)$ and $\overline{\mathcal{EO}}(n)$ denote the paper's parity-separated partition-counting functions, and let $p_0(n)$ and $p_1(n)$ be the associated partition functions. Inequality conjecture.

(i) For every $n\geq3$,

$$\overline{\mathcal{OE}}(2n+1)>\overline{\mathcal{EO}}(2n).$$

(ii) For every $n\geq1$,

$$p_0(n)\geq p_1(n),$$

with strict inequality unless $n\in\{1,10,13\}$. These inequalities conjecturally describe a systematic dominance between the two parity-separated partition families; the supplied text gives no resolution status beyond presenting them among the paper's conjectures.