$\mu$. Then $X$ has the CSWP.

By Corollary 5.4 of [@KU2], every such ${\mathrm{supt}}(\mu)$ is separable in the case that $X$ is $\aleph_0$–dissipated; for a LOTS $X$, this was a much earlier folklore result.

\[cor-CSWP-sep\] If $X$ fails the CSWP and is $\aleph_0$–dissipated, then some compact separable subspace of $X$ fails the CSWP.

Is there a compact space $X$ which fails the CSWP such that all compact separable subspaces of $X$ satisfy the CSWP?

This $X$ cannot be one of the three examples already known to fail the CSWP — namely, any space containing either the Cantor set [@RUD1] or $\beta{{\mathbb N}}$ [@HS] or the examples of [@HK2; @HK3] (obtained assuming $\diamondsuit$ or CH), since all these spaces are separable.

Now, considering products of LOTSes:

\[lemma-prod-cswp\] Assume that $X$ is a compact subset of $\prod_{\alpha < \kappa} L_\alpha$, where each $L_\alpha$ is a LOTS, and assume that $X$ does not have the CSWP. Then for some separable closed compact $H_\alpha \subseteq L_\alpha$, the space $X \cap \prod_{\alpha < \kappa} H_\alpha$ also fails the CSWP.

Let $\pi_\alpha : X \to L_\alpha$ be the usual coordinate projection. We may assume that each $L_\alpha = \pi_\alpha(X)$, so that $L_\alpha$ is compact. Fix $\mu$ 