Lapid–Mao's local identity for Whittaker-period constants

Let $F$ be the base number field, let $E/F$ be the quadratic extension, let $v$ be a place of $F$, and let $\Pi_v$ be the relevant local base-change representation. Write $\omega_{\Pi_v}$ for its central character, let $\eta\in E_v^\times$ satisfy $\bar\eta=-\eta$, and let $c_{\pi_v}$ be the local constant occurring in the preceding global formula.

Lapid–Mao's local conjecture. For every $\eta\in E_v^\times$ such that $\bar\eta=-\eta$, one has

$$c_{\pi_v}=\omega_{\Pi_v}(\eta).$$

In particular, the source states that this local identity implies the preceding global Whittaker-period conjecture for $\pi$. The supplied text does not specify whether the local identity has been resolved.