Polynomial-volume equidistribution conjecture for periodic torus orbits

Fix $\rho > 0$. Let $G=\mathbf{G}(\mathbb{R})$ be an $\mathbb{R}$-split real algebraic group, let $\Gamma < G$ be an arithmetic lattice, and let $H$ be a maximal $\mathbb{R}$-split torus. Let $x_iH$ be a sequence of periodic $H$-orbits satisfying

$$\operatorname{vol}(x_iH) \geq \operatorname{disc}(x_iH)^\rho.$$

A weak limit of the measures $\mu_{x_iH}$ is called algebraic when it is an algebraic measure. Polynomial-volume equidistribution conjecture. Any weak limit of the measures $\mu_{x_iH}$ is algebraic. The conjecture asserts that the small-volume periodic orbits responsible for known nonequidistribution are the only obstruction; the source gives no resolution.