Type-and-Jordan-type invariance conjecture for unitary stable orbital integral strata

Let $L$ be a unimodular hermitian lattice over the ring of integers $\boldsymbol{o}_E$ of an unramified quadratic extension $E/F$, and let $M,M'$ be sublattices of $L$. Write $\mathcal{SO}_{\gamma,M}$ for the volume of the stratum associated with $M$, and suppose that $M$ and $M'$ have the same type and the same Jordan type. Type-and-Jordan-type invariance conjecture. Then

$$\mathcal{SO}_{\gamma,M}=\mathcal{SO}_{\gamma,M'}.$$

This asserts that these two discrete invariants determine the stratum volume in the unitary case. The supplied text gives no resolution status or further evidence.