Factoring maps to big Cohen-Macaulay algebras through blowups

We show that for a reduced universally catenary equidimensional Noetherian local ring $R$, any sufficiently functorial big Cohen-Macaulay algebra assignment $R \mapsto B_R$, and any proper birational map $Y \to \mathrm{Spec} R$, there exists a factorization $R \to {\bf R}\Gamma(Y, \mathcal{O}_Y) \to B_R$ in the derived category of $R$-modules.