Tightness conjecture for the centered maximum of the two-dimensional discrete Gaussian free field

Let $V_N=([0,N]\cap\mathbb{Z})^2$ and let $\{X_z^N\}_{z\in V_N}$ be the zero-mean discrete two-dimensional Gaussian free field on $V_N$ with Dirichlet boundary conditions. Define

$$X_N^*=\max_{z\in V_N}X_z^N,\qquad Y_N:=X_N^*-E X_N^*.$$

Tightness conjecture. The sequence of random variables $\{Y_N\}_{N\geq 1}$ is tight.

The paper proves tightness along a dense deterministic subsequence, while tightness for the full sequence remains open. The conjecture concerns the fluctuations of the maximum after centering by its mean, beyond the established first-order asymptotics $X_N^*/\log N\to 2\sqrt{2/\pi}$.