Stratum-wise constancy conjecture for multiplicities in complex-rank parabolic category O

Let $\scr{X}_n$ be the parameter space with strata $W_I$, and let $W_I^{\natural}\subset W_I$ consist of points $(\bt,\bs)$ whose coordinates $t_i$ are not integers. A point of $W_I$ is $W_I$-generic when its Galois orbit is Zariski dense in $W_I$. Multiplicity constancy conjecture. For a stratum $W_I$ with $\dim W_I\geq 1$, the multiplicity $[\mts(\blambda):\lts(\bmu)]$ is independent of the choice of $(\bt,\bs)\in W_I^{\natural}$ and can therefore be computed at any $W_I$-generic point. If $W_I$ is interpolatable, these multiplicities can be computed through the stated ultraproduct multiplicity result. This conjecture predicts that multiplicities are controlled by the stratification of parameter space, extending the available computation at interpolatable strata; its status is not resolved in the supplied text.