Let $A$ be a ring, $I\subset A$ a finitely generated ideal.

The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit of étale ring maps that are the identity on mod $I$ fibers:

$$A^h = \varinjlim_{s\in S} A_s$$

where $A\to A_s$ is étale and such that $A/I\to A_s/I$ is the identity, and $S$ is an index set.

When $A$ is smooth over the Noetherian henselian valuation ring $R = \mathbf{Z}_{(p)}^h$ and $I = pA$ is principal, can $S$ be arranged to be a countable set?