Monotonicity and uniqueness conjecture for the smallest eigenvalue

Let $\Lambda_\kappa$ denote the smallest eigenvalue of the relevant weighted spherical operator, with parameter $\kappa>0$ in dimension $n$. Proposition~ gives three possible alternatives for the eigenspace associated with $\Lambda_\kappa$. Monotonicity and uniqueness conjecture. (i) When $\kappa>0$ and $n\geqslant3$, only statement (ii) of Proposition~ is true. (ii) The function $\kappa\mapsto\Lambda_\kappa$ is increasing. This conjecture refines the proposition and is based on numerical experiments; its validity is not established in the supplied text.