Distribution conjecture for Hasse failures on Markoff-type cubic surfaces

For each integer $k\geq 0$, let $\mathfrak{h}(k)$ be the associated class number, and call $k$ admissible when it satisfies the paper's admissibility condition. For $K\geq 0$, let $C_0>0$, let $C_t>0$ for each $t\geq 1$, and let $\theta$ satisfy

$$\frac{1}{2}<\theta<1.$$

Distribution conjecture. The number of Hasse failures for $0\leq k\leq K$ satisfies

$$\left|\left\{0\leq k\leq K:\mathfrak{h}(k)=0\ \text{and}\ k\ \text{admissible}\right\}\right|\sim C_0K^{\theta}.$$

More generally, for $t\geq 1$,

$$\left|\left\{0\leq k\leq K:\mathfrak{h}(k)=t\right\}\right|\sim C_tK^{\theta}.$$

This conjecture predicts a power-law distribution for the number of Hasse failures and, more generally, for parameters having any fixed positive class number. The source gives numerical motivation but no resolution.