Refinement conjecture for the secant degeneracy length of binary-form strata

Let $\mu\vdash d$ be a partition, let $\bar\mu$ denote the associated standard stratum, and let $\ell_{\bar\mu}$ and $\ell_{\mu'}$ denote the corresponding secant degeneracy lengths. Write $\mu'\succeq\mu$ for the refinement partial order. Refinement conjecture. For every partition $\mu\vdash d$, there exists a partition $\mu'\succeq\mu$ such that

$$\ell_{\bar\mu}=\ell_{\mu'}.$$

The claim is based on extensive calculations and appears in the paper's final list of open problems; no resolution is supplied in the source.