Dirichlet disk eigenfunction growth conjecture for $1\leq\gamma<3$

Let $\{\lambda_{n_l,m_l}\}$ be a strictly increasing subsequence of Dirichlet Laplace eigenvalues on the unit disk, with associated $L^2$-normalized eigenfunctions $F_{n_l,m_l}$. Define

$$\frac{\log m_l}{\log n_l}=\gamma_l,$$

and suppose $\gamma_l\to\gamma$. Set

$$\phi(\gamma)=\liminf_{l\to\infty}\frac{\log\lVert F_{n_l,m_l}\rVert_\infty}{\log\lambda_{n_l,m_l}}.$$

Dirichlet disk growth conjecture. If $1\leq\gamma<3$, then

$$\phi(\gamma)=\frac{1}{4}-\frac{1}{6\gamma}.$$

The conjecture is motivated by numerical simulations. The paper proves only the lower bound $\phi(\gamma)\geq\frac{1}{12}$ for $\gamma\geq1$, so the asserted formula remains open in the stated range.