Przebinda's character formula conjecture for theta lifting

Let $R_s(MG_1,MG_2)$ denote the relevant stable-range representation class, let $\theta_s(MG_1,MG_2)(\pi)$ be the stable theta lift, and let $\omega^c$ be the contragredient oscillator representation. Theta-lift Hom-space conjecture. For every $\pi\in R_s(MG_1,MG_2)$,

$$V(\theta_s(MG_1,MG_2)(\pi))\cong [\operatorname{Hom}_{\mathfrak g_1,MK_1}(\omega^c,\pi)]_{MK_2}.$$

The paper presents this as a critical step in Przebinda's program to recover the Harish-Chandra character of a theta lift from that of the original representation; it is not established there.