The $1/4$ subdiffusivity exponent conjecture for simple random walk on the UIPQ

Let $Q_\infty$ be the uniform infinite planar quadrangulation (UIPQ), rooted at $\rho$, and let $(X_k)_{k\geq 0}$ be simple random walk on $Q_\infty$ started at $\rho$. Write $\mathrm{d_{gr}}$ for graph distance. The walk is called subdiffusive if its displacement grows more slowly than the diffusive scale $n^{1/2}$. The $1/4$ subdiffusivity conjecture. The subdiffusivity exponent of the simple random walk on the UIPQ is $1/4$:

$$\max_{0 \leq k \leq n}\mathrm{d_{gr}}(\rho,X_k) \approx n^{1/4}.$$

The paper proves the upper bound $n^{1/3}$ up to logarithmic factors and conjectures that this exponent is not sharp; the precise meaning of $\approx$ and the correct logarithmic fluctuations remain to be established.