Standard conjecture for nearby and vanishing cohomology

Let $X\to S$ be the projective family in the paper, let $X_s$ be its special fiber, let $X_\eta$ be its generic fiber, and let $H^*_{\psi}(X_s)$ and $H^*_{\phi}(X_s)$ denote the nearby- and vanishing-cycle cohomology groups. Fix an ample line bundle $L$ on $X$, and let ${\mathsf L}$ be cup product with $c_1(L)\in H^2(X_s)(1)$. Set $n=\dim X_\eta$. Standard conjecture for nearby and vanishing cohomology. For $i\leq n$, there is an isomorphism

$${\mathsf L}^i:H^{n-i}_{\psi}(X_s)\overset{\sim}{\longrightarrow}H^{n+i}_{\psi}(X_s)(i),$$

and for $i\leq n+1$, there is an isomorphism

$${\mathsf L}^i:H^{n+1-i}_{\phi}(X_s)\overset{\sim}{\longrightarrow}H^{n+1+i}_{\phi}(X_s)(i).$$

This is proposed as an analogue of Grothendieck's standard conjectures for nearby and vanishing cohomology; the supplied text gives no resolution status.