Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\mathbf{A}^1,tr}\colon Sh_{Nis}^{tr} \to HI_{Nis}^{tr}$ and $h_0^{\mathbf{A}^1}\colon Sh_{Nis}\to HI_{Nis}$ be the two localizations to the hearts of the homotopy t-structures of $DM^{eff}$ and $DA^{eff}$. For $F\in Sh_{Nis}^{tr}$, there is a canonical surjective map $h_0^{\mathbf{A}^1}(u(F))\to u(h_0^{\mathbf{A^1},tr}(F))$.
Is it known whether or not this map is an isomorphism? In general the left hand side is very difficult to compute while the right hand side is Suslin homology.
