Equal-area subspectrality conjecture for rectangles

Let $R$ be a rectangle. For $\epsilon>0$, let $R_\epsilon$ be the rectangle obtained from $R$ by the map

$$(x,y)\mapsto(\epsilon x,y/\epsilon).$$

Equal-area subspectral rectangles conjecture. There exists $s>0$ such that, for every $0<\epsilon<s$, $R_\epsilon$ is Dirichlet-superspectral to $R$ and Neumann-subspectral to $R$. This numerical conjecture concerns area-preserving deformations of rectangles and is presented as open in the source context.