Bounded maximum under positivity conditioning for the membrane model

Let $n=2$ or $n=3$, let $V_N$ denote the finite domain of the membrane model, let $\psi=(\psi_x)_{x\in V_N}$ be its field, let $\operatorname{E}_N$ denote expectation, and let $\Omega_{V_N,+}=\\{\psi_x\geq 0\text{ for all }x\in V_N\\}$. Bounded-maximum conjecture. We conjecture that

$$\lim_{N\to\infty}\operatorname{E}_N\left(N^{-\frac{4-n}{2}}\max_{x\in V_N}\psi_x\mid\Omega_{V_N,+}\right)<\infty.$$

For conditioning on positivity in a smaller domain, the maximum is already known to remain of order $N^{\frac{4-n}{2}}$ and there is no entropic repulsion; the conjecture asserts the analogous bounded-order statement when the whole domain is constrained, where the conditioning event has exponentially small probability and the proof is more difficult. The authors further expect a typical conditioned field to be of order $c d_N(x)^{\frac{4-n}{2}}$, but that expectation is not part of the asserted conjecture.