Neumann disk eigenfunction growth conjecture for $\gamma\geq1$

Let $F^N_{n,m}$ be the $L^2$-normalized eigenfunctions of the Neumann Laplace operator on the unit disk, with eigenvalues $\lambda^N_{n,m}$. For a subsequence with $\gamma_l=\frac{\log m_l}{\log n_l}\to\gamma$, define $\phi^N(\gamma)$ as in the Dirichlet case, replacing $F_{n,m}$ and $\lambda_{n,m}$ by $F^N_{n,m}$ and $\lambda^N_{n,m}$. Neumann disk growth conjecture. If $\gamma\geq1$, then

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

The conjecture is based on numerical simulations. The paper establishes only the lower bound $\phi^N(\gamma)\geq\frac{1}{12}$ for $\gamma\geq1$ and states that no precise formula was obtained for $\gamma>3$, so the conjecture remains open.