Rank-condition conjecture for configuration varieties of diagrams

Let $D$ be a diagram whose columns are subsets of $\\{1,\ldots,N\\}$, and let $\mathcal F_D$ be its configuration variety, consisting of configurations $(V_C)_{C\in D}$ of subspaces with $\dim V_C=|C|$. For every list of columns $C,C',\ldots$ of $D$, impose the rank conditions

$$\dim(V_C+V_{C'}+\cdots)\leq |C\cup C'\cup\cdots|,$$ $$\dim(V_C\cap V_{C'}\cap\cdots)\geq |C\cap C'\cap\cdots|.$$

Rank-condition conjecture. For an arbitrary diagram $D$, $\mathcal F_D$ is the set of configurations satisfying these conditions for every list of columns $C,C',\ldots$ of $D$. Equivalently, the variety defined by these equations is irreducible.

The preceding proposition establishes this description for northwest diagrams. The conjecture asks for the analogous rank-condition description, and irreducibility, for arbitrary diagrams; the source gives no resolution.