Proposition: Canonical agreement checks the restricted inclusion only

Let $r_O$ denote restriction to $O=U_{\mathrm{can}}$ and $Y=\ker r_O$. If $\tilde{\mathsf{M}}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ agrees with the canonical contraction on $O$, i.e.\ $r_O\tilde{\mathsf{M}}=m^{\circ }\mathsf{Rec}$, then for every $n\in \ker \mathsf{Rec}$, $r_O\tilde{\mathsf{M}}(n)=0$, i.e.\ $\tilde{\mathsf{M}}(\ker \mathsf{Rec})\subseteq Y$. This does not decide whether $\tilde{\mathsf{M}}(n)=0$.

Proof. $r_O\tilde{\mathsf{M}}(n)=m^{\circ }\mathsf{Rec}(n)=0$. A distribution supported in $W\setminus O$ is not detected by $r_O$; this is the last paragraph of XIV-J's proof of the descent criterion Jdescent.