Monomial positivity conjecture for products of Schur functions

Let $Y_n$ denote the set of Young diagrams with $n$ boxes, and let $s_\lambda$ be the Schur symmetric function indexed by $\lambda$. Let $\lambda,\hat\lambda\in Y_n$ and $\mu,\hat\mu\in Y_{n-1}$ be pairs of Young diagrams such that both pairs differ by moving the box $(i,j)$ to $(\hat i,\hat j)$, where $\hat i>i$, and suppose that

$$\lambda\setminus\mu=\hat\lambda\setminus\hat\mu=(r,c).$$

A symmetric function is monomial-positive when all coefficients in its monomial expansion are nonnegative. Monomial positivity conjecture. If $r<i$, then $s_\lambda s_{\hat\mu}-s_{\hat\lambda}s_\mu$ is monomial-positive. If $r>\hat i$, then $s_{\hat\lambda}s_\mu-s_\lambda s_{\hat\mu}$ is monomial-positive. The source motivates this as a generalization of an elementary monotonicity proposition; related quadratic expressions have been studied, but this form is stated as apparently new and remains unresolved in the supplied text.