Let $(R, \mathfrak m,k)$ be a local Gorenstein ring of dimension $2$ such that $\mu (\mathfrak m^2)(=\dim_k \mathfrak m^2/\mathfrak m^3) =3$ .
Then is it true that $R$ is regular ? Or at least is it true that $R$ has minimal multiplicity i.e. $e(R)=\mu(\mathfrak m)-1$ ?
here $e(R)=d! \lim_{n \to \infty} \dfrac {l (R/\mathfrak m^n)}{n^d}=(d-1)! \lim_{n \to \infty} \dfrac {\mu(\mathfrak m^n)}{n^{d-1}} $ where $d=\dim R$ , so in our case $e(R)=2 \lim_{n \to \infty} \dfrac {l (R/\mathfrak m^n)}{n^2}=\lim_{n \to \infty} \dfrac {\mu(\mathfrak m^n)}{n}$.
If need be, I'm willing to assume $k$ is infinite and/or $R$ contains a field
Note that if $k$ is infinite, then a necessary and sufficient condition for a local Cohen-Macaulay ring to have minimal multiplicity is that $\mathfrak m^2=\mathfrak m (\overline x)$ for some regular sequence $\overline x$ in $R$