Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal.

We know that for any smooth $R/I$-algebra $A_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A_0$.

We also know that for any map $A_0\to B_0$ of smooth $R/I$-algebras, there exist $R$-smooth algebras $A$ and $B$, and a map $A\to B$ that lifts $A_0\to B_0$.

Suppose $A_0\to B_0$ is surjective. Can $A\to B$ be arranged to be surjective too?