Recursive tetris-type formula for powers of the Vandermonde determinant

Let $1\leq m\leq n$. Let $\lambda\vdash n(n-1)$ satisfy $n-1\leq\ell(\lambda)\leq n$ and

$$\lambda_1=\lambda_2=\cdots=\lambda_m=2n-m-2.$$

Let $\mu\vdash n(n+1)$ satisfy

$$\mu_1=2n-m,\qquad \mu_2=\cdots=\mu_{m+1}=2n-m-1,$$

and, if $m<n$, $\mu_j=\lambda_{j-1}$ for $j=m+2,\ldots,n+1$. Here $a_{\delta_n}$ denotes the Vandermonde determinant associated with the staircase partition $\delta_n$, and $s_\lambda$ denotes the Schur function indexed by $\lambda$; $\langle\,\ \rangle$ is the relevant Schur-function inner product. Recursive tetris-type conjecture. One has

$$\left\langle a_{\delta_{n+1}}^{2},s_\mu\right\rangle=(-1)^m(m+1)\left\langle a_{\delta_n}^{2},s_\lambda\right\rangle.$$

This conjectural recursion describes how a specified tetris-type modification of the partition changes the Schur coefficient in the square of the Vandermonde determinant. The authors report verification for $n\leq 6$ using Maple; the general statement remains open.