Conjecture on character differences associated with Foulkes' conjecture

Let $\rho^{(a)}_b$ and $\chi^{(r)}$ denote the characters used in the source, let $\boxtimes$ denote the indicated outer product, and let $\operatorname{Char}(S_N)$ be the cone of genuine characters of $S_N$. For integers $1\le a\le b$, consider the following virtual characters of $S_{ab-1}$. Character-positivity conjecture.

(i)

$$\rho^{(a)}_{b-1}\boxtimes \chi^{(a-1)}-\rho^{(a-1)}_b\boxtimes \chi^{(b-1)}\in\operatorname{Char}(S_{ab-1});$$

(ii)

$$\frac{\rho^{(b)}_a-\rho^{(a)}_b}{\chi^{(1)}}\in\operatorname{Char}(S_{ab-1}).$$

The conjecture is based on computational data in small cases and is motivated by Foulkes' conjecture, which predicts positivity of $\rho_a^{(b)}-\rho_b^{(a)}$ in $\operatorname{Char}(S_{ab})$. Its resolution is not specified in the source.