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This part only shows the existence of lower bounds, which is not the point. See edit. This is consistently true for example when the near coherence principle of ultrafilters holds. It says, for any two non-principal ultrafilters $U, V$ there exists a finite-to-one $f: \omega\to \omega$ such that $f(U)=f(V)$. Since $f$ is finite-to-one, it makes sure $f(U)=f(V)$ is not principal.

Edit: the answer is consistently no. Suppose there exist two non-isomorphic Ramsey ultrafilters $U, V$, then we claim $U\cdot V$ and $V\cdot U$ do not have an infimum. This follows from the following facts

1. $U\cdot V, V\cdot U$ are neither P-point nor Q-point
2. $U\cdot V \simeq V\cdot U$ implies $U\simeq V$
3. The only elements that are potentially RK below $U\cdot V$ are $V\cdot U$, $U$, $V$ (or the principal ultrafilters)
4. So the common non-principal lower bounds for $U\cdot V, V\cdot U$ are $U, V$. But they are incomparable.

All of these can be found in Blass' thesis.