Limits of graded Gorenstein algebras of Hilbert function $(1,3^k,1)$

Let $R=\mathsf{k}[x,y,z]$, the polynomial ring over a field $\mathsf{k}$. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field. We here show that when $\mathsf{k}$ is algebraically closed, and the Hilbert function sequence $T=(1,3^k,1),k\ge 2$ (i.e. $T=(1,3,3,\dots ,3,1)$ where $k$ is the multiplicity of $3$) then the family $G_T$ parametrizing graded Artinian algebra quotients $A=R\mathrm{/}I$ of $R$ having Hilbert function $T$ is irreducible, and $G_T$ is the closure of the family $\mathrm{G}\mathrm{o}\mathrm{r}(T)$ of Artinian Gorenstein algebras of Hilbert function $T$. We then classify up to isomorphism the elements of these families $\mathrm{G}\mathrm{o}\mathrm{r}(T)$ and of $G_T$. Finally, we give examples of codimension three Gorenstein sequences, such as $(1,3,5,3,1)$, for which $G_T$ has several irreducible components, one being the Zariski closure of $\mathrm{G}\mathrm{o}\mathrm{r}(T)$.