Self-duality conjecture for neural-collapse solutions

Consider the unconstrained feature model with classifier matrix $\boldsymbol{W}$ and feature matrix $\boldsymbol{H}$. Let $(\boldsymbol{W}^*,\boldsymbol{H}^*)$ be a global solution and let $\boldsymbol{h}_k^*$ denote the class feature aggregate.

Self-duality conjecture. The class feature aggregate is proportional to its corresponding classifier weight: for each $k=1,\ldots,K$, $\boldsymbol{h}_k^*=\alpha\boldsymbol{w}_k^*$ for some $\alpha>0$ independent of $k$.

This conjecture expresses the self-duality expected between classifier weights and class features in neural collapse. Its resolution is not specified in the source.