Let ${\mathbb{𝔸}}_{n,m}$ be the polynomial ring $\text{Sym}({ℂ}^n\otimes {ℂ}^m)$ with the natural action of ${\text{GL}}_m(ℂ)$. We consider a family of ${\text{GL}}_m(ℂ)$-stable ideals $J_{n,m}$ in ${\mathbb{𝔸}}_{n,m}$, each equivariantly generated by one homogeneous polynomial of degree $2$ and show that the regularity of this family is unbounded. Using this, we negatively answer a question raised by Erman, Sam and Snowden on a generalization of Stillman’s conjecture.