The si-full-rank conjecture for Deligne–Lusztig cohomology maps

Let $w=sw's$ be an element of the Weyl group with $ll(w)=ll(w')+2$, and let $Z=X(w)\cup X(sw')$. For $i\geq 0$, write

$$H^{i}_c(Z)=H^{i-2}_c(Z')(-1)=A\oplus B,$$

where

$$A\cong \operatorname{coker}\big(H^{i-3}_c(X(w'))\longrightarrow H^{i-2}_c(X(w's))\big)(-1)$$

and

$$B=\ker\big(H^{i-2}_c(X(w'))\longrightarrow H^{i-1}_c(X(w's))\big)(-1).$$

Let $r^i=r^i_{w,sw'}:H^i_c(Z)\longrightarrow H^i_c(X(sw'))$ be the natural map. Si-full-rank conjecture. For $i\geq 0$, the restriction

$$r^i_{\mid B}:B\longrightarrow H^i_c(X(sw'))$$

has si-full rank. The conjecture is motivated by the preceding surjectivity result and concerns the contribution of the summand $B$ to the compactly supported cohomology of the Deligne–Lusztig variety $X(sw')$; its status is not resolved in the supplied source.