The conjecture that the affine Hecke cocenter maps isomorphically to the p-adic cocenter

Let ${\bf H}=S_c^\infty(G)$ be the algebra of locally constant compactly supported functions on the $p$-adic group, let ${\bf J}$ be the corresponding algebra, and let $\Phi:{\bf H}\to {\bf J}$ be the homomorphism whose restriction to Iwahori bi-invariant vectors recovers the homomorphism $\phi$. For an algebra $A$, write its cocenter as $A/[A,A]$. Cocenter isomorphism conjecture. The map $\Phi$ induces an isomorphism of cocenters.

This conjectural extension would generalize the cocenter isomorphism theorem for the affine Hecke algebra to the full algebra of locally constant compactly supported functions on the $p$-adic group. The source indicates that injectivity is expected to follow from density of tempered characters, but does not establish the claim.