The toric M-variety conjecture for homology with closed support

A real algebraic variety is a complex variety $X$ defined over the real numbers, with real part $X(\mathbb{R})$ equal to the fixed-point set of complex conjugation. It is an M-variety when

$$\sum_k b_k(X(\mathbb{R}))=\sum_k b_k(X(\mathbb{C})).$$

For homology with closed support, the toric M-variety conjecture. Every toric variety is an M-variety.

Nonsingular complete toric varieties and algebraic tori are known to be M-varieties, but the conjecture concerns arbitrary toric varieties and homology with closed support.