Joint equidistribution conjecture for primitive restriction forms

Let $Q$ be a positive definite integral quadratic form, let $L$ be a rational $k$-dimensional subspace, and let $q_L$ and $q_{L^\perp}$ be the associated integral quadratic forms obtained by restricting $Q$ to the relevant lattices. For an integral quadratic form $q$, let $\gcd(q)$ be the largest positive integer dividing it and write $\widetilde q=\gcd(q)^{-1}q$ for its primitive part. Joint primitive-form equidistribution conjecture. If $(L_i)_{i\in\mathbb{N}}$ is a sequence of rational subspaces with $\operatorname{disc}_Q(L_i)\to\infty$, then

$$\det(\widetilde q_{L_i},\mathbb{Z}^k)\quad\text{and}\quad\det(\widetilde q_{L_i^\perp},\mathbb{Z}^{n-k})\to\infty.$$

This is a technical growth condition intended to control the primitive parts of the restriction forms and is proved in the paper when $k\neq n-k$, while the equal-rank case remains open.