Four-dimensional subdiffusive heat-kernel decay conjecture

Let $d=4$, and let $\mathbb P$ be any i.i.d. environment law satisfying $\mathbb P(0<\omega_b\leq 1)=1$. Write $\cmss P_\omega^{2n}(0,0)$ for the quenched return probability at time $2n$ in the environment $\omega$. Four-dimensional subdiffusive heat-kernel decay conjecture. For every such environment law,

$$\frac{n^2}{\log n}\cmss P_\omega^{2n}(0,0)\xrightarrow[n \to \infty]{}0,$$

$\mathbb P$-almost surely. This would show that the logarithmic correction in dimension four cannot be attained along an entire sequence in any fixed environment, complementing the theorem giving arbitrarily close lower bounds along a subsequence. The conjecture was proved in an upcoming preprint by Biskup, Louidor, Rozinov and Vandenberg-Rodes.