What's the easiest (by which I mean uses the least fancy machinery) proof of the direct summand conjecture in dimension 2?

Recall that the direct summand conjecture says that:

**Conjecture** (Hochster): If $R$ is a regular ring and $S$ is a module finite integral extension, then $R \to S$ splits as a map of $R$-modules.

It is trivial in characteristic zero (via the trace map) and not that hard in characteristic $p > 0$ using Frobenius-type methods. In mixed characteristic it is known up to dimension 3.