Görtz–Haines–Kottwitz–Reuman conjecture for affine Deligne–Lusztig varieties

Let $G=Sp_4(L)$, where $L$ is as in the paper. For $bin G(L)$, let

$$J_b=\{g\in G(L)\mid gbsigma(g)^{-1}=b\}$$

be the $\sigma$-centralizer of $b$, and let $\operatorname{def}(b)$ be the difference between the $F$-rank of $G$ and the $F$-rank of $J_b$. Let $b'\in G(L)$ be basic with $\kappa(b)=\kappa(b')$, and let $\tilde{w}\in\widetilde{W}$ have sufficiently large length. Görtz–Haines–Kottwitz–Reuman conjecture. One should have

$$X_{\tilde{w}}(b)\neq\emptyset\quad\Longleftrightarrow\quad X_{\tilde{w}}(b')\neq\emptyset.$$

When these varieties are nonempty, one should also have

$$\dim X_{\tilde{w}}(b)=\dim X_{\tilde{w}}(b')-\langle\nu_b,\rho\rangle+\frac{1}{2}\bigl(\operatorname{def}(b')-\operatorname{def}(b)\bigr).$$

The paper states that this conjecture is true for $G=Sp_4(L)$, so the claim is solved in the setting considered here.