The $G^1$-degeneration conjecture for real toric varieties

Let $X$ be a toric variety with real part $X(\mathbb{R})$. Let $G^k_{p,q}$ denote the spectral sequence for $H_*(X(\mathbb{R}))$ arising from the real toric filtration.

The $G^1$-degeneration conjecture. The spectral sequence $G^k_{p,q}$ for $H_*(X(\mathbb{R}))$ degenerates at the $G^1$ level.

This conjecture sharpens the earlier conjecture about degeneration of the complex toric homology spectral sequence. The comparison results identify the $G^1$ and $E^2$ terms as graded groups, but the corresponding spectral-sequence differentials are not compatible, so degeneration at $G^1$ remains a separate open assertion.