In short, the question is
What do we know about the sheaf $\pi_*\underline{\bar{\mathbb{Q}}_{\ell}}$ given by the family of (very original, see below) Hessenberg varieties for $GL_n$? As a sum of shifted perverse sheaves, does every component have full support?
Fix a choice of Borel $B\subset GL_n$ (over any field of preference, say $\mathbb{C}$), and $\mathfrak{b}=\operatorname{Lie} B$. Let $\mathfrak{h}\subset\mathfrak{gl}_n$ be the sum of $\mathfrak{b}$ and the negative simple root subspaces. Consider the family of Hessenberg varieties $$\tilde{\mathfrak{g}}_{\mathfrak{h}}:=\{(g,\gamma)\in GL_n/B\times\mathfrak{gl}_n\;|\;g^{-1}\gamma g\in\mathfrak{h}\}$$ with the natural map $\pi:\tilde{\mathfrak{g}}_{\mathfrak{h}}\twoheadrightarrow\mathfrak{gl}_n$ by $(g,\gamma)\mapsto\gamma$. The fibers of $\pi$ are the very original, so-called Hessenberg varieties. They are paved by affine spaces, see e.g. [1]. The generic fiber is the toric variety given by the Weyl chamber decompositions (where torus is the centralizer in $PGL_n$) [2, $\S$IV].
[1] Tymoczko, Julianna, Linear conditions imposed on flag varieties, Am. J. Math. 128, No. 6, 1587-1604 (2006). ZBL1106.14038.
[2] De Mari, F.; Procesi, C.; Shayman, M. A., Hessenberg varieties, Trans. Am. Math. Soc. 332, No. 2, 529-534 (1992). ZBL0770.14022.
