$R$ is $F$-coherent. Quite generally, we note that if $R \to S \to T$ is a composition of ring homomorphisms such that $R \to T$ is étale and $R \to S$ is unramified, then $S \to T$ is étale. In particular, it is flat.

Finally, since the flat colimit of coherent rings is coherent, the henselization of $R$ can be constructed as a colimit of various localizations of module-finite étale $R$-algebras, we conclude that $R^{\mathrm{h}}$ is $F$-coherent.

It is worth suggesting the following list of questions for prompting the research in future.

Let $R \to S$ be a faithfully flat map of Noetherian rings such that $R$ is $F$-coherent. Then what conditions on the fibers of $R \to S$ are required in order that $S$ is $F$-coherent?

We do not know if this question is true for the case where $S$ is smooth over $R$, because we do not even know whether the polynomial ring $R[x]$ is $F$-coherent or not, assuming that $R$ is $F$-coherent.

Let $R$ be a Noetherian ring and let $t \in R$ be a non-zero divisor. If $R/tR$ is $F$-coherent, then is $R$ also $F$-coherent?

This question is not obvious, since the property of coherence is 