The critical-dimension crossover conjecture for parabolic Anderson valleys

Let $u$ be the solution of the parabolic Anderson equation with multiplicative noise, started from bounded initial data $u_0$. For $\beta$ as in the definition of the macroscopic Hausdorff dimension $\mathrm{Dim}_{\mathds{H}}$ and the associated set functional $\mathcal{S}_{\beta}$, consider the spacetime set of points where $u(t,t^{2/3}x)$ is below the critical valley threshold. Critical-dimension crossover conjecture. There is a constant $C>0$, depending on $\beta$ and $u_0$, such that, almost surely,

\mathrm{Dim}_{\mathds{H}}\Big(\mathcal{S}_{\beta}\Big(\big\\{(t,x)\in (e,\infty)\times \mathds{R}: u(t,t^{2/3}x)\leq e^{-\frac{t}{24} -\alpha (t\log \log t)^{1/3}}\big\\}\Big)\Big)=2-C\alpha^3

for every $\alpha\in [0,C^{-1/3}]$. The conjecture predicts a crossover of the macroscopic dimension at the critical value $\gamma=1/24$, extending the dimension formula to the case excluded by the main theorem; the constant and the asserted almost-sure dimension remain to be established.