Ghosh–Sarnak conjecture on integral Hasse failures for Markoff surfaces

Let $U_m$ be the affine Markoff surface

$$U_m:\ x^2+y^2+z^2-xyz=m,$$

where $m$ is an integer, and let $U_m(\mathbb{A}_{\mathbb{Z}})$ denote its integral adelic points. Ghosh–Sarnak conjecture. The number of integral Hasse failures satisfies

$$\#\{m\in\mathbb{Z}:0\leq m\leq M,\ U_m(\mathbb{A}_{\mathbb{Z}})\neq\emptyset\ \text{but}\ U_m(\mathbb{Z})=\emptyset\}\approx C_0M^\theta,$$

for some $C_0>0$ and some $\frac{1}{2}<\theta<1$. This conjecture predicts that integral Hasse failures occur with an order of magnitude strictly between $M^{1/2}$ and $M$; the supplied source states that it is disproved, while earlier work had established only lower bounds for failures not explained by the Brauer–Manin obstruction.