Relative trace formula comparison for BZSV periods

Let $Q=LU$ be a parabolic subgroup of $G$, let $N_L$ be a maximal unipotent subgroup of $L$, set $N=N_LU$, and let $\xi_L$ be a generic character of $[N_L]$, extended trivially across $U$. For Schwartz functions $f$ on $G(\mathbb A)$ and $f'$ on $L(\mathbb A)$, let $I(f)$ be the relative trace formula using the $\Delta$-period and the $(N,\xi_L)$-period, and let $J(f')$ be the relative trace formula using the $\widehat{(\hat{\Delta}_{\mathrm{red}})}$-period and the $(N_L,\xi_L)$-period. Relative trace formula comparison. There should be a character $\chi$ of $L$, depending on $\Delta$, such that

$$I(f)=J(\mathcal F_\chi(f'))$$

for all $f\in\mathcal S(G(\mathbb A))$ that is locally $L$-supercuspidal. This comparison is proposed as a way to relate the two period problems and is known in the source only in selected special cases, including some polarized examples.