  We resolve an open question posed by the authors of arXiv:1501.06925v2 in
2015 concerning a Gr\"obner theoretic approach for the noetherianity of the
twisted commutative algebra ${\rm Sym}({\rm Sym}^2(\mathbf{C}^\infty))$. We
provide a negative answer to their question by producing an explicit antichain.
In doing so, we establish a connection to well studied posets of graphs under
the subgraph and induced subgraph relation. We then analyze this connection to
suggest future paths of investigation, for example a deeper study of initial
ideals in twisted commutative algebras.
