Conjectured hitting-probability threshold for slow-growth points

Let $K\subset\mathbb{R}$ be a nonrandom compact set, let $\mathfrak S(\theta)$ be the random set of points of slow growth at parameter $\theta$, and let $\lambda(\theta)$ denote the associated function. Write $\operatorname{\dim_{_{\rm H}}}K$ for the Hausdorff dimension of $K$. Hitting-probability threshold conjecture. The probability that $\mathfrak S(\theta)$ intersects $K$ should satisfy

$$\mathrm{P}\{\mathfrak S(\theta)\cap K\neq\varnothing\}=\begin{cases} 1 & \text{if $\lambda(\theta) < \frac12\operatorname{\dim_{_{\rm H}}} K$},\\ 0 & \text{if $\lambda(\theta) > \frac12\operatorname{\dim_{_{\rm H}}} K$}. \end{cases}$$

This is motivated by the analogous hitting-probability threshold for Brownian slow points. The source presents it as a belief and does not provide a resolution, so it remains open.