The linearity conjecture for support loci of Ext modules

Let $cM$ be a regular holonomic $cD_X$-module, and let $W\to X$ be a relative compact open subset. Suppose that, locally on $W$,

$$cExt^l_{cD_{X,R}}(cN_k,cD_{X,R})\not=0$$

for some $n\le l\le n+r$. For a support locus $S_j(\mathscr Ext^l_{\mathscr D_{X,R}}(\mathscr N_k,\mathscr D_{X,R}))$, the linearity conjecture. If it is nonempty, then it is a finite union of translated $(j-n)$-codimensional linear subspaces of $\mathbb C^r$. If $\mathcal M$ underlies a $\mathbb Q$-mixed Hodge module, for example $\mathcal M=\mathscr O_X$, then all these support loci are defined over $\mathbb Q$. The conjecture is motivated by the relative codimension filtration and known results on regular holonomic maximal extensions; the source notes that it holds for $l=n$, while the cases with $n<l\le n+r$ remain conjectural.