Conjecture on the emptiness of the critical slow-point set

Let $\theta_c$ be the critical parameter and let $\mathfrak S(\theta_c)$ denote the corresponding random set of points of slow growth. Write $\operatorname{dim_{_{\rm H}}}$ for Hausdorff dimension. Critical slow-point emptiness conjecture.

$$\mathfrak S(\theta_c)=\varnothing \quad\text{a.s.}$$

This is the parabolic SPDE analogue of the corresponding assertion for slow points of Brownian motion. The best result currently available is only $\operatorname{\dim_{_{\rm H}}}\mathfrak S(\theta_c)=0$, so the conjecture remains open.