Ghosh–Sarnak conjecture on the number of Hasse failures for Markoff surfaces

For each integer $m$, let $\mathcal{U}_m$ denote the corresponding integral Markoff surface, and let $\mathbb{A}_{\mathbb{Z}}$ denote the adelic space used to formulate the integral Hasse principle. Ghosh–Sarnak's conjecture. The number of integers $m$ with $0\leq m\leq M$ for which $\mathcal{U}_m$ has adelic integral points but no integral point satisfies

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

for some $C_0>0$ and some $\frac12<\theta<1$. The conjecture concerns the asymptotic frequency of integral Hasse failures. The supplied status evidence says that it is disproved; consequently, the asserted asymptotic is not correct as stated.