# Flatness of certain subrings

The following question appears, more or less, here:

Let $$k$$ be an algebraically closed field of characteristic zero and let $$S$$ be a commutative $$k$$-algebra (I do not mind to further assume that $$S$$ is an integral domain).

Let $$R_1,R_2 \subseteq S$$ be two (probably different) $$k$$-subalgebras of $$S$$ such that $$R_i \subseteq S$$, $$1 \leq i \leq 2$$ is: finitely generated, flat and separable (in other words, $$R_i \subseteq S$$ is etale).

Denote the $$k$$-algebra generated by $$R_1$$ and $$R_2$$ by $$R$$.

Is it true that $$R \subseteq S$$ is flat?

Clearly, $$R \subseteq S$$ is finitely generated and separable (hence unramified). But what about flatness? I guess (?) that there exists a counterexample in dimension two.

A non-counterexample: $$R_1=k[x^2], R_2=k[x^3], S=k[x]$$; here $$R=k[x^2,x^3] \subseteq k[x]$$ is not flat, but also $$R_i \subseteq S$$ are not separable (althouh they are finitely generated and free). See this question, which explains that $$k[x^2,x^3] \subseteq k[x]$$ is not separable, hence $$k[x^2] \subseteq k[x]$$ and $$k[x^3] \subseteq k[x]$$ are not separable, since if at least one of them were, then $$k[x^2,x^3] \subseteq k[x]$$ was separable, by a result concerning separability).

Any hints and comments are welcome; thank you!