Another question on compact set decomposition

Question

I previously asked (see Compact refinement of a covering) about whether there is a compact set covered by 2 open sets which cannot be decomposed as the union of 2 (not necessarily disjoint) compact subsets, each of which is contained in one of the open sets. That compactification of $\mathbb{Q}$ solved that question. Now I'm curious whether there is such a compact set if you add the condition that the space must be locally compact. (This is asking for another counterexample to the previous question which is additionally locally compact.) By locally compact I mean every point has a neighborhood base of compact sets.

Answer 1

This is not an answer, but it is too long for a comment and shows some restrictions if we search for a counterexample (which, as I believe, exists).

Known facts:

(1) For compact $T_1$-spaces, Hausdorff is equivalent to normal.

(2) For $KC$-spaces, the "compact refinement property" implies normal. The converse holds for compact $KC$-spaces.

Thus, any non-Hausdorff compact $KC$-space is a counterexample to your first question. But now we have

(3) For locally compact spaces, $KC$ is equivalent to Hausdorff.

The implications Hausdorff $\Rightarrow$ $KC$ $\Rightarrow$ $T_1$ are true without any further assumptions. So let $X$ be a locally compact (which means that each point has a neighborhood base consisting of compact sets) $KC$-space. Consider two distinct points $x,y \in X$. There exist an open neighborhhod $U$ of $x$ and a compact $C$ such that $U \subset C \subset X \setminus \{ y \}$. But $C$ is closed, hence $V = X \setminus C$ is an open neighborhood of $y$ such that $U \cap V = \emptyset$.

This means that we cannot find a counterexample among compact locally compact $KC$-spaces.

Here is some related material:

Is a locally compact space a KC-space if and only if it is Hausdorff?

Künzi, Hans-Peter A., and Dominic van der Zypen. "The Construction of Finer Compact Topologies." Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2005. https://www.researchgate.net/publication/30814746_The_Construction_of_Finer_Compact_Topologies