Monotonicity conjecture for plethysm coefficients with a quadratic Schur function

Let $\nu$, $\lambda$, and $\lambda$ be partitions, with $\nu$ and $\lambda$ arbitrary. Write $\nu\sqcup(1)$ for the partition obtained by adjoining a part $1$ to $\nu$, and similarly write $(\lambda+(1))\sqcup(1)$ for the corresponding partition operation. For partitions $\nu$ and $\lambda$, let $\langle s_\nu\circ s_{(2)}\mid s_\lambda\rangle$ denote the coefficient of $s_\lambda$ in the plethysm $s_\nu\circ s_{(2)}$. Monotonicity conjecture. For arbitrary partitions $\nu$ and $\lambda$, one has

$$\langle s_\nu\circ s_{(2)}\mid s_\lambda\rangle\leqslant\langle s_{\nu\sqcup(1)}\circ s_{(2)}\mid s_{(\lambda+(1))\sqcup(1)}\rangle.$$

This proposed monotonicity property concerns plethysm coefficients and is presented as a new property motivated by known monotonicity phenomena in plethysm. Its status is not established in the supplied text.