Witness-elimination conjecture for rectangularly complementary Schur products

Fix a rectangle $R$. Let $P_0$ be the set of products $s_\lambda s_{\lambda^{\mathrm c}}$ for complementary pairs in $R$. Recursively define $W_i$ to consist of those products in $P_i$ for which there is a partition $\pi$ such that $s_\pi$ occurs in that product and in no other product in $P_i$, and set $P_{i+1}=P_i\setminus W_i$.

Witness-elimination conjecture. With

$$P_0=\{s_\lambda s_{\lambda^{\mathrm c}}\},$$ $$W_i=\{s_\lambda s_{\lambda^{\mathrm c}}\in P_i\mid \exists\pi:\ s_\lambda s_{\lambda^{\mathrm c}}\text{ is the only product in }P_i\text{ containing }s_\pi\},$$ $$P_{i+1}=P_i\setminus W_i,$$

there is an index $i$ for which $P_i$ is empty.

The conjecture strengthens linear independence by requiring that every product can be successively eliminated using a Schur-function witness. It is presented as a natural extension of the witness construction proving the self-complementary case; no resolution is given in the source.