Zilber dichotomy for $\mathrm{DCF}_{0,m}$

We prove that the theory of differentially closed fields of characteristic zero in $m\geq 1$ commuting derivations DCF$_{0,m}$ satisfies the expected form of the dichotomy. Namely, any minimal type is either locally modular or nonorthogonal to the (algebraically closed) field of constants. This dichotomy is well known for finite-dimensional types; however, a proof that includes the possible case of infinite dimension does not explicitly appear elsewhere.