Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $M$.

Question: Is there a norm one Banach space projection $\Phi: B(L^{2}(M,\tau))\rightarrow R$ such that $\Phi|_M=\mathbb{E}_{R}$, where $\mathbb{E}_{R}$ is the $\tau$-preserving conditional expectation of $M$ onto $R$?

(I actually expect this is never possible, since by a result of Tomiyama the map $\Phi$ is necessarily singular, but have been unable to prove this. I figured this is well-known, so a reference would be a fine answer!)

Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $M$.

Question: Is there a norm one Banach space projection $\Phi: B(L^{2}(M,\tau))\rightarrow R$ such that $\Phi|_M=\mathbb{E}_{R}$, where $\mathbb{E}_{R}$ is the $\tau$-preserving conditional expectation of $M$ onto $R$?

(I actually expect this is never possible, but have been unable to prove this. I figured this should be well-known, so a reference would be a fine answer!)