Parity of parabolic induction from 0-cuspidal pairs

Let $G$ be a reductive group, let $P$ be a parabolic subgroup of $G$, and let $L$ be a Levi subgroup of $P$. Let $(C,\mathcal{E})\in\mathscr{I}(L)^{0-\operatorname{cusp}}$, where $\mathscr{I}(L)^{0-\operatorname{cusp}}$ denotes the set of $0$-cuspidal pairs on $L$. Write $\operatorname{Ind}^G_P$ for parabolic induction and $\mathcal{IC}(C,\mathcal{E})$ for the corresponding intersection cohomology complex.

Parity induction conjecture. The complex

$$\operatorname{Ind}^G_P\mathcal{IC}(C,\mathcal{E})$$

is a parity complex.

The claim is stated in the context of parity sheaves in equivariant settings. The supplied text gives no resolution or further evidence concerning its status.