Conjecture on vanishing of Picard–Lefschetz pairings modulo the identity component

In the situation of the Picard–Lefschetz formula, let $c_1=w_1c_0$ and $c_2=w_2c_0$ with $w_1,w_2\in W$, and let $W_\chi^0$ be the specified subgroup of $W$. Assume that the cosets $w_1W_\chi^0$ and $w_2W_\chi^0$ are distinct as elements of $W/W_\chi^0$. Vanishing conjecture. Then

$$\partial u_1\pitchfork\partial u_2^*=0.$$

This is explicitly described as a slight modification of a preceding corollary. The parser supplies no evidence resolving it, so its status remains open.