A poset $\mathbb{P}$ is called *well-met* iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.

Question: Suppose $\mathbb{P}$ is a separative partial order which is $\lambda$-directed closed (for some regular infinite cardinal $\lambda$). Can we always view $\mathbb{P}$ as a dense suborder of a well-met poset which is still $\lambda$-directed closed?

I'm vaguely aware that Boolean completions can screw up properties like directed closure, but I'm only asking for finite infima, not arbitrary infima. It's not clear to me that the obvious "well-met closure" of $\mathbb{P}$ is still $\lambda$-directed closed, but I also don't have a counterexample.

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