Asymptotic minimization of the first Dirichlet eigenvalue in a square with two hard obstacles

We study an asymptotic minimization problem for the first Dirichlet eigenvalue in the square $Q=(-1,1)^2$, where two equal hard circular obstacles of radius $r$ move inside $Q$ with disjoint interiors; tangency between the obstacles and with the outer boundary is allowed. We prove that, as $r\to0$,