A counterexample to a Gr\"obner approach for noetherianity of the twisted commutative algebra ${\rm Sym}({\rm Sym}^2(\mathbf{C}^\infty))$

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.