The complete type-edge-pair description of the solution set for $Sp_4$

Let $S$ be the solution set for the relevant affine Deligne–Lusztig problem for $Sp_4(L)$. For every type-edge pair $(t,e)$, let $S_{(t,e)}$ be the collection of final chambers of possible foldings of the associated composite galleries, and define

$$S_1=\bigcup_{(t,e)}S_{(t,e)}.$$

Complete type-edge-pair conjecture. The union over all type-edge pairs equals the solution set:

$$\bigcup_{(t,e)}S_{(t,e)}=S,$$

or, equivalently, $S_1=S$.

The assertion is proposed for the cases $b\neq 1$ after a conjectural superset has been constructed. For $b=1$, the paper states that the corresponding superset and subset computations already establish equality; for other $b$, the full verification and construction of a matching subset remain undone.