\[thA\] and \[thB\], Question \[Q1\] is affirmative for $R$ with $bg(R)\leq 2$. Question \[Q2\] has positive answer given in Proposition \[p1\] if $bg(R)\leq 1$, but there is the following counter-example, even when $bg(R)=2$.

Let $R$ be the numerical semigroup ring $k[[H]]$, where $H=\langle 3,13,14\rangle$. Then $\ell_R(R/\omega^*(\omega))=\mathsf{res}(H)=4$ by [@HHS Remark 7.14]. there is, however, a Gorenstein subring $S=k[[H']]$ of $R$, where $H'=\langle 6,9,13,14,16,17\rangle$. The colength $\ell_S(R/S)$ is equal to $2$, and so this gives a counter-example of Question \[Q2\].

We now return to prove the Theorem \[thB\].

\(2) $\implies$ (1): This is a consequence of Lemma \[lem23\] by letting $I=\mathfrak{a}$.

\(1) $\implies$ (2): In the case $bg(R)\leq 1$, assertion is followed by Theorem \[thA\]. So we may assume $bg(R)=2$. Take a Gorenstein local ring $(S,\mathfrak{n})$ and module-finite residually rational birational extension $S\subset R$ satisfying $\ell_S(R/S)=2$.

Let $B$ be the ring $\mathfrak{n}:\mathfrak{n}$. By Lemma \[lem22\] and Lemma \[lem21\], we have $\ell_S(B/S)=1$ and $S\subsetneq B\subseteq R$. Therefore $\ell_B(R/B)=1$ and $B$ is local. Let $\m_B$ be the maximal ideal of $B$ and fix a preimage $t\in R$ of a basis $\overline{t}$ of the one-dimensional vector space $R/B$ over $B/\m_B$. By the relation $\m_Bt\subseteq B$ yields that $t\in B:\m_B=\m_B:\m_B$. Therefore $R=B+Bt\subseteq \m_B:\m_B$. In particular, $R\m_B\subseteq 