Hermitian Kazhdan–Lusztig polynomial sign conjecture

Let $X=X({\mathcal O},{\mathcal L})$ be a standard module for the affine Hecke algebra with Hermitian Kazhdan–Lusztig polynomials $P^h_{({\mathcal O},{\mathcal L}),({\mathcal O}',{\mathcal L}')}(q)$ and ordinary Kazhdan–Lusztig polynomials $P_{({\mathcal O},{\mathcal L}),({\mathcal O}',{\mathcal L}')}(q)$. Hermitian Kazhdan–Lusztig sign conjecture. For every pair $({\mathcal O},{\mathcal L})$, there exists an orientation number $\epsilon({\mathcal O},{\mathcal L})\in\{\pm1\}$ such that

$$P^h_{({\mathcal O},{\mathcal L}),({\mathcal O}',{\mathcal L}')}(q)=\epsilon({\mathcal O},{\mathcal L})\epsilon({\mathcal O}',{\mathcal L}')P_{({\mathcal O},{\mathcal L}),({\mathcal O}',{\mathcal L}')}(-q).$$

The conjecture predicts that Hermitian Kazhdan–Lusztig polynomials are obtained from the ordinary ones by replacing $q$ with $-q$, up to signs attached to the two orbit–local-system pairs; the supplied context says it is motivated by a theorem of Adams, van Leeuwen, Trapa, and Vogan but gives no resolution.