Every separable metric space $A$ has a metrizable compactification, i.e. a compact metrizable space $X$ for which $A$ embeds topologically as a dense subspace of $X$. There are many approaches to constructing metrizable compactifications and probably several I am unaware of.

**Question:** Given an arbitrary separable metrizable space $A$, does there exist a metrizable compactification $X$ of $A$ such that there does not exist any path $\alpha:[0,1]\to X$ with $\alpha(0)\in A$ and $\alpha(1)\in X\setminus A$?

In other words, I'd like a metrizable compactification of $A$ so that every path component of $A$ is also a path component of the compactification. I can answer this question affirmatively when $A$ is locally compact (using a topologist's-sine modification of the one-point metric compactification) but this doesn't generalize and I only have ad-hoc approaches for non-locally compact cases.