A conjecture on the action of Hecke operators

Let $F$ be a local non-archimedian field, $L$ be a central division $F$-algebra of rank $n$ and ${\mathcal{A}}_n(L)$ be the convolution algebra of smooth compactly supported $Ad$-invariant complex-valued measures on $L^{\ast }$. It is known that for different division $F$-algebras $L$ of rank $n$ the algebras ${\mathcal{A}}_n(L)$ are canonically isomorphic. In this paper I propose a conjecture extending these isomorphisms to the algebras generated by Hecke operators on spaces of $1\mathrm{/}2$-measures on the stacks of $L^{\ast }$-bundles on smooth complete curves over $F$.