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 $Y\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$, there exists a factorization $R\to \mathbf{R}\mathrm{\Gamma }(Y,{\mathcal{O}}_Y)\to B_R$ in the derived category of $R$-modules.