Conjecture on minimizers and maximizers of the generalized eigenvalue quotient

For a domain $\Omega$, let $\lambda_1(p;\Omega)$ denote its first $p$-Laplacian eigenvalue, and define

$$\mathcal{R}_{p,q}(\Omega)=\frac{\lambda_1(p;\Omega)^{\frac{1}{p}}}{\lambda_1(q;\Omega)^{\frac{1}{q}}}.$$

Generalized eigenvalue quotient conjecture. If $q<p$, then functionals of the form $\mathcal{R}_{p,q}(\Omega)$ admit a minimizer and are bounded above. Any sequence of fixed-volume sets $\Omega_k$ satisfying

$$\operatorname{diam}\Omega_k\to\infty$$

is a maximizing sequence; the supremum is not attained and equals

$$\frac{\lambda_1(p;I)^{\frac{1}{p}}}{\lambda_1(q;I)^{\frac{1}{q}}},$$

where $I\subset\mathbb{R}$ is an interval. This extends the paper's reverse Cheeger-type questions to quotients of first $p$- and $q$-Laplacian eigenvalues; the stated minimization, boundedness, maximizing-sequence characterization, and nonattainment remain conjectural in the cited passage.