Critical-noise-level conjecture for full intermittency

Let $U\subset\mathbb{R}^d$ be a bounded domain satisfying the hypotheses of either Theorem~ or Theorem~, and let $u(t,x)$ be the corresponding solution to the stochastic heat equation with Dirichlet boundary condition. Let $\lambda_0$ and $\lambda_1$ be the thresholds defined in those theorems, so that $\lambda_0\leq\lambda_1$. Critical-noise-level conjecture. There exists $\lambda^*\in[\lambda_0,\lambda_1]$ such that, when $\lambda>\lambda^*$, the solution $u(t,x)$ is fully intermittent, while, when $\lambda<\lambda^*$, all its $p$-th moments for $p\geq2$ are bounded in time and it is not fully intermittent. The moment estimates in the two theorems leave the intermediate regime unresolved; the conjecture predicts a sharp transition at a critical noise level, but no resolution is given here.