Realizability conjecture for universally pseudoeffective surface classes

Let $X$ be a smooth complex projective variety, and set

$$\mathrm{H}^{1,1}(X)_{\mathbb{R}}\coloneq \mathrm{H}^2(X,\mathbb{R})\cap\mathrm{H}^{1,1}(X).$$

Let $\mathrm{B}_2(X)$ denote the cone of universally pseudoeffective $2$-dimensional classes, and let a class be realizable over $\mathbb{R}$ when it is represented in the sense defined in the paper by real algebraic or geometric cycles. Realizability conjecture. For every universally pseudoeffective $\eta\in\mathrm{B}_2(X)$, the following are equivalent: $\eta$ is realizable over $\mathbb{R}$, and the bilinear form

$$\mathrm{H}^{1,1}(X)_{\mathbb{R}}\times\mathrm{H}^{1,1}(X)_{\mathbb{R}}\longrightarrow\mathbb{R},\qquad (\alpha,\beta)\longmapsto\int_{\eta}\alpha\wedge\beta$$

has at most one positive eigenvalue. The conjecture proposes a numerical description of the realizable cone inside the cone of universally pseudoeffective classes; the surrounding results establish related cases and implications, but the general equivalence remains open.