cannot apply Remark for this choice of $S$.

Let $S'=k[[t^6,t^9,t^{10}, t^{13},t^{14}]]$ and $\mathfrak{n}'$ be its maximal ideal. Then $S'$ is Gorenstein and $S'\subseteq R$ is module-finite residually rational birational extension with $\ell_{S'}(R/S')=3$. Put $B'=\mathfrak{n}':\mathfrak{n}'=k[[t^6,t^9,t^{10},t^{13},t^{14},t^{17}]]$. Then $\m_{B'}:\m_{B'}$ is equal to $k[[t^3,t^4]]$, which contains $R$. By Remark, $R$ has a self-dual ideal $I=\m_{B'}$ with $\ell_R(R/I)=3$. Consequently, Question \[Q1\] is affirmative for $R$ regardless of whether $bg(R)=2$ or not.

The seif-duality of the maximal ideal
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In this section, we collect some properties of local rings $(R,\m)$ with $\m\cong \m^\dag$.

Let $(R,\m)$ be a Cohen-Macaulay local ring with a canonical module. Assume $\m\cong \m^\dag$. Then

-   $\dim R\leq 1$.

-   Let $x\in\m\setminus \m^2$ be a non-zero divisor of $R$. Then $R/(x)$ also has self-dual maximal ideal.

-   $\edim(R)=r(R)+1$.

Suppose $\dim R\geq 2$ and $\omega$ is a canonical module of $R$. Applying $(-)^\dag$ to the exact sequence $0 \to \m \to R \to k \to 0$, we get an exact sequence $$0 \to \Hom_R(k,\omega) \to \m^\dag \to \omega \to \Ext_R^1(k,\omega).$$ By the assumption $\dim R\geq 2$ yields that $\Hom_R(k,\omega)=0=\Ext_R^1(k,\omega)$ and hence $\m^\dag \cong R$. From the isomorphism $\m\cong \m^\dag$, it follows that $\m$ is principal ideal. This shows that $\dim R\leq 1$, which is 