Sup-norm conjecture for simultaneous eigenvectors of supersingular isogeny graphs

Let $A_\ell$ denote the weighted adjacency matrix of the supersingular $\ell$-isogeny graph $\mathcal{G}(p,\ell)$ for all $\ell\neq p$. Let $v$ be an $L^2$-normalized simultaneous eigenvector of the operators $A_\ell$. Sup-norm conjecture. One has

$$\|v\|_\infty \;\ll\; \frac{\log p}{\sqrt{p}}.$$

This is evidence toward the strongest expected scale $p^{-1/2+\varepsilon}$ for sup-norms, which would express complete delocalization of eigenvectors. The stated logarithmic bound is presented as supporting evidence in the regular case $N=1$; the general optimal sup-norm bound is described as open.