Duality compatibility conjecture for the filtered nearby-cycle object

Let $\Psi$ be the nearby-cycle object with filtrations $F$ and $G$, let $\mathbb{D}$ denote duality, and let $b$ be the isomorphism relating the dual nearby cycles to $\Psi(-1-\delta_X)$. For indices $p,q$, the graded pieces are identified with direct sums over strata $D_I$ satisfying $\lvert I\rvert=p+q+1$. Duality compatibility conjecture. The following diagram is commutative:

\xymatrix{ \mathbb{D}(\operatorname{Gr}^F_p\operatorname{Gr}_G^q \Psi) \ar[r]^{\sim} \ar[d]_{\sim} & \operatorname{Gr}_G^p\operatorname{Gr}^F_q \mathbb{D}(\Psi) \ar[r]^{\sim} & \operatorname{Gr}_G^p \operatorname{Gr}^F_q \Psi(-1-\delta_X) \ar[d]_{\sim}\\ \displaystyle \bigoplus_{\lvert I\rvert=p+q+1} \mathbb{D}(\mathbb{R}\underline{\Gamma}^{\dagger}_{D_I}(\mathcal{O}_{\mathfrak{X},\mathbb{Q}})(q+1)[p+q+1]) \ar[rr]^{\sim} & & \displaystyle \bigoplus_{\lvert I\rvert=p+q+1} \mathbb{R}\underline{\Gamma}^{\dagger}_{D_I}(\mathcal{O}_{\mathfrak{X},\mathbb{Q}})(p-\delta_X)[p+q+1] }

where the bottom isomorphism is induced by

$$\mathbb{D}(\mathbb{R}\underline{\Gamma}^{\dagger}_{D_I}(\mathcal{O}_{\mathfrak{X},\mathbb{Q}})[p+q+1])\simeq\mathbb{R}\underline{\Gamma}^{\dagger}_{D_I}(\mathcal{O}_{\mathfrak{X},\mathbb{Q}})(-\delta_X+p+q+1)[p+q+1].$$

The source states this as a conjectural commutativity assertion in the duality discussion; no resolution status is supplied.