Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$. Assume that:

**(1)** $R$ and $S$ are (Noetherian) integral domains.

**(2)** $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull dimension.

**(3)** $R$ is regular (hence a UFD).

**(4)** $S$ is Cohen-Macaulay.

**(5)** $R \subseteq S$ is simple, namely, $S=R[w]$ for some $w \in S$.

**(6)** $R \subseteq S$ is free.

**(7)** $R \subseteq S$ is integral, namely, every $s \in S$ satisfies a monic polynomial over $R$.

**(8)** $m_RS=m_S$, namely, the extension of $m_R$ to $S$ is $m_S$.

**(9)** It is not known whether the fields of fractions of $R$ and $S$, $Q(R)$ and $Q(S)$, are equal or not.

**(10)** It is not known if $R \subseteq S$ is separable or not.

**Remark:** It is known that if a (commutative) integral domains ring extension $A \subseteq B$ is integral+flat, then it is faithfully flat, and if also $Q(A)=Q(B)$, then $A=B$.
This is why I did not want to assume that $Q(R)=Q(S)$, since in this case $R=S$ immediately.

Question:Is it true that, assuming(1)-(10)imply that $S$ is regular or $R=S$?

**Example:**
$R=\mathbb{C}[x(x-1)]_{x(x-1)}$ and $S=\mathbb{C}[x]_{(x)}$,
with $R \neq S$ and $S$ is regular.

**Non-example:** $R=\mathbb{C}[x^2]_{(x^2)}$ and $S=\mathbb{C}[x^2,x^3]_{(x^2,x^3)}$, but condition **(8)** is not satisfied.

Relevant questions, for example: a, b, c, d.

Thank you very much! I have asked the above question here, with no comments (yet).