Minimax exclusivity conjecture for power-type losses

Let $p,q>0$ be distinct exponents, and for each $r>0$ let $\mathcal{L}_r$ be the class of losses whose local behavior near $a=\theta$ is $c(\theta)|\theta-a|^r+o(|\theta-a|^r)$ with $c(\theta)>0$. Consider minimax optimality over a regular parametric model. Minimax exclusivity conjecture. No estimator is simultaneously minimax for a loss in $\mathcal{L}_p$ and a loss in $\mathcal{L}_q$. Equivalently, the family $\{\mathcal{L}_r:r>0\}$ forms a minimax exclusivity partition of $\mathscr{L}_{\mathrm{power}}=\bigcup_{r>0}\mathcal{L}_r$. The structural classes are disjoint and closed cones under positive scaling, but whether their structural separation forces minimax exclusivity remains unclear.