The $a$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let $A$ be a regular local ring and let $\mathcal{F} = \{F_{n}\}_{n \in \mathbb{Z} }$ be a filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus _{n \geq 0}F_{n}$ is a Noetherian ring with $\mathrm{dim} \mathcal{R}(\mathcal{F}) = \mathrm{dim} A + 1$. Let $\mathcal{G}(\mathcal{F}) = \bigoplus _{n \geq 0}F_{n}/F_{n+1}$ and let $\mathrm{a}(\mathcal{G}(\mathcal{F}))$ be the $a$-invariant of $\mathcal{G}(\mathcal{F})$. Then the theorem says that $F_{1}$ is a principal ideal and $F_{n} = F_{1}^{n}$ for all $n \in \mathbb{Z}$ if and only if $\mathcal{G}(\mathcal{F})$ is a Gorenstein ring and $\mathrm{a}(\mathcal{G}(\mathcal{F})) = -1$. Hence $\mathrm{a}(\mathcal{G}(\mathcal{F})) \leq -2$, if $\mathcal{G}(\mathcal{F})$is a Gorenstein ring, but the ideal $F_{1}$ is not principal.