Let $(S, \mathfrak n)$ be a regular local ring. For $0\ne f\in \mathfrak n^2$ define $c(f, S):=\{\text{ideals } I \text{ of } S : f\in I^2\}$ .
Now let $(S_1, \mathfrak n_2)$ and $(S_2,\mathfrak n_2)$ be two regular local rings with the same residue field and let $0\ne f_i\in \mathfrak n_i^2, i=1,2$ be such that $S_1/(f_1)\cong S_2/(f_2)$ (isomorphic as local rings https://stacks.math.columbia.edu/tag/07BJ )
My question is: If $c(f_1, S_1)$ is a finite set, then is it true that so is $c(f_2,S_2)$ ?
(If needed, I'm willing to assume that $S_1,S_2$ and $S_i/(f_i)$ are complete)