Stratification conjecture for quasiderivation fiber dimensions

Let $G/P$ be a homogeneous space, let $M_\lambda$ be the maximal proper $P$-submodule associated with a parameter $\lambda$, and let $Z_\lambda\subset\mathfrak{g}/\mathfrak{p}\times M_\lambda$ be the corresponding incidence variety. Denote by $\pi$ the projection of $Z_\lambda$ to $\mathfrak{g}/\mathfrak{p}$ and set $l_\lambda(x)=\dim\pi^{-1}(x)$. Stratification conjecture. There exists an algebraic stratification

$$\mathfrak{g}/\mathfrak{p}=\mathop{\sqcup}\limits_{i=1}^rX_i$$

such that for any $\lambda$ the function $l_\lambda(x)$ is constant along each $X_i$:

$$l_\lambda(x)=l_\lambda^i,\quad x\in X_i.$$

This asserts uniform geometric control of the incidence fibers arising in the study of zeros of generic sections of homogeneous vector bundles. The supplied text gives no resolution or further evidence for the conjecture.