The codimension-two spin-genus representation conjecture for quadratic forms

Let $q:\mathbb{Z}^m\to\mathbb{Z}$ and $Q:\mathbb{Z}^n\to\mathbb{Z}$ be integral quadratic forms with $m<n$, and let

$$\min(q):=\min_{x\in\mathbb{Z}^m\setminus\{0\}}q(x).$$

Assume that $q$ is primitively represented by an element of the spin genus of $Q$, and that $n-m=2$. Codimension-two spin-genus representation conjecture. There exists a constant $C(Q)>0$ such that

$$\min(q)\geq C(Q)$$

implies that $q$ is primitively represented by $Q$. The local representation and minimum conditions are insufficient in codimension two because of spinor obstructions; the conjecture addresses whether the spin-genus condition becomes sufficient for sufficiently large minimum. Its $m=1$ case, concerning primitive representations of integers, has been established by Duke and Schulze-Pillot using estimates for Fourier coefficients of half-integral-weight forms, so the general statement is resolved only in that special case.