Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$-space

Yes, in ZFC. The paper proves a stronger statement: if a Tychonoff space $X$ is not a $P$-space, then $C_p(X)$ contains a closed, separable, metrizable linear subspace that is not pseudocomplete. A closed separable subspace is the closure of a countable subset, so if all such closures are pseudocomplete, $X$ must be a $P$-space. Dow and Tkachuk had shown this under the extra assumption that $X$ is functionally countable (their Theorem 3.10); the result removes that assumption. The companion Question 4.9 (whether $X$ must also be functionally countable) is not addressed.

Yes, in ZFC. The paper proves a stronger statement: if a Tychonoff space $X$ is not a $P$-space, then $C_p(X)$ contains a closed, separable, metrizable linear subspace that is not pseudocomplete. A closed separable subspace is the closure of a countable subset, so if all such closures are pseudocomplete, $X$ must be a $P$-space. Dow and Tkachuk had shown this under the extra assumption that $X$ is functionally countable (their Theorem 3.10); the result removes that assumption. The companion Question 4.9 (whether $X$ must also be functionally countable) is not addressed.