Symmetry conjecture for toric periods

Let $\phi\in\mathcal{S}_N^\mathrm{new}(\mathcal{O})$ be a normalized Hecke eigenform, let $\pi$ be the cuspidal representation generated by $\phi$, and let $Y(D;\pi)$ be the set of quadratic fields satisfying the local root-number condition. Let $\mathfrak{o}_\pi$ be the ring of integers of the Hecke field, and define

$$\mathbb{P}_x[\mathfrak{P}(\phi)=z]=\frac{\#\{E\in Y(D;\pi)\mid |\Delta_E|<x,\ \mathfrak{P}_E(\phi)=z\}}{\#\{E\in Y(D;\pi)\mid |\Delta_E|<x\}}.$$

Symmetry conjecture. For $x>0$ and sufficiently small $\delta>0$,

$$\frac12\sum_{z\in\mathfrak{o}_\pi}\left|\mathbb{P}_x[\mathfrak{P}(\phi)=z]-\mathbb{P}_x[\mathfrak{P}(\phi)=-z]\right|\ll\frac{x^{1-\delta}}{\#\{E\in Y(D;\pi)\mid |\Delta_E|<x\}}.$$

This predicts asymptotic sign symmetry in the distribution of algebraic toric periods and is supported by numerical experiments; it remains open.