Grochow's determinantal ideal closure conjecture

Let $X$ be an $n\times n$ matrix of indeterminates, and let $I_n$ be the ideal generated by the $\tfrac{n}{2}\times\tfrac{n}{2}$ minors of $X$. For a nonzero polynomial $f\in I_n$, an $f$-oracle circuit is an algebraic circuit allowed to use gates computing $f$. Grochow's determinantal ideal closure conjecture. For every nonzero polynomial $f\in I_n$, there is a constant-depth algebraic circuit of size $\operatorname{poly}(n)$ with $f$-oracle gates that computes the $m\times m$ determinant for some $m=n^{\Theta(1)}$. Grochow's conjecture is a non-principal-ideal analogue of closure under factorization. The paper's results establish related debordering results but retain polynomial dependence on $\deg(f)$; removing that dependence is left as an open question.