In the lecture notes on condensed mathematics the solidification of the free condensed abelian group $\mathbb{Z}[S]$ on a profinite set $S$ is defined as the inverse limit $\lim_{\leftarrow} \mathbb{Z}[S_i]$, but it can also alternatively be described as $$ \mathbb{Z}[S]^\blacksquare \cong \underline{\mathrm{Hom}}(C(S,\mathbb{Z}), \mathbb{Z}). $$ My question is whether this also holds for general compact Hausdorff spaces $S$. In the case that $X$ is a CW complex this seems to follow from example 6.5 where $\mathbb{Z}[X]^{\blacksquare} \cong H_\bullet(X)$ is shown.

This also seems to follow from it's derived analogue, which was stated in part 2 of the answer here, if we can show that $\mathbb{Z}[X]$ is pseudo-coherent for any compact Hausdorff space $X$. I think this holds because we can resolve $X$ by extremally disconnected spaces, which should yield a resolution of $\mathbb{Z}[X]$ by compact projectives.

By analogy I also have the same question for the $\mathcal{M}$-completeness of condensed $\mathbb{R}$-vector spaces. Is it true that the condensed vector space of signed Radon measures on a compact Hausdorff space $X$ can be given as: $$ \mathcal{M}(X) \cong \underline{\mathrm{Hom}}_{\mathbb{R}}( C( X, \mathbb{R}) , \mathbb{R}) $$ Again it seems like all the hints are in the lecture notes, but I'm not comfortable enough with the subject to pin down the details.

A short note on why I'm asking this: the main reason is that I want to better understand the analogy with measure theory where one says that a condensed $\mathbb{R}$-vector space $V$ is $\mathcal{M}$-complete if for any continuous $f: K \to A$ and measure $\mu$ on $K$ one can form the "integral" $\int f\ d\mu$. I would find this much more convincing if I can let $K$ be a finite CW-complex rather than just a profinite set.

On this note I also have a third, optional question: if the above two descriptions work, is there something similar one can say in the case of $\mathcal{M}_p(S)$ and for $p$-liquid vector spaces?