Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A_0\to B_0$ a finite faithfully flat lci map of smooth $R_0 := R/I$-algebras.

We fix a smooth $R$-algebra $A$ lifting $A_0$ and assume $A$ and $A_0$ integral.

Does there exist a smooth $R$-algebra $B$ with a finite $R$-map $A\to B$?

If a finite $R$-map $A\to B$ exists, then it is finite lci and faithfully flat, because it is an integral extension and so it is a finite surjective lci on spectra.

I'm trying to use this and variations on the theme. I've been able to show that there is a quasi-finite syntomic map $A\to B$ with $B$ not necessarily smooth.