Rigidity for a semilinear Neumann problem with exponential nonlinearity in the large diffusion limit

We consider a semilinear Neumann problem with exponential nonlinearity in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$. We prove that there exists a threshold $\bar{\epsilon }>0$ such that for all $\epsilon >\bar{\epsilon }$, any classical solution must be constant. This result provides a positive answer to a conjecture recently posed by Calanchi, Ciraolo, and Messina (2026). Our proof relies on a combination of $L^1$-estimates, a Jensen-type argument via the Neumann Green’s function to obtain uniform exponential integrability, and elliptic regularity.