Linear independence of rectangularly complementary Schur products

Let $R$ be a rectangle, and let $u$ and $u^{\mathrm c}$ range over unordered complementary pairs of partitions in $R$. In the ring $\Lambda$ of symmetric formal polynomials, write $s_\nu s_{\nu^{\mathrm c}}$ for the corresponding product of Schur functions.

Linear-independence conjecture. Fix a rectangle $R$. The products

$$s_\lambda s_{\lambda^{\mathrm c}}$$

are all linearly independent.

The theorem in the paper proves this for the self-complementary products, while the conjecture asks for the full family of complementary pairs. The source also proposes a stronger witness-elimination statement, which implies this conjecture.