The integrality and zero-block conjecture for quadratic lattice extensions

Let $L=M\oplus N$ be a quadratic lattice, where $M$ is spanned by $(e_1,\ldots,e_{n-d})$ and $N$ by $(e_{n-d+1},\ldots,e_n)$. Let $L'=M'\oplus N$ be a lattice containing $L$ in $L\otimes_{\mathfrak{o}}F$, with $M'$ containing $M$ in $M\otimes_{\mathfrak{o}}F$. Let $q_{L'}$ and $q_{M'}$ be the quadratic forms induced from $q_L$. When $q_{M'}$ is integral, write $\bar{M}'=M'/\pi M'$ for its reduction modulo $\pi$; when $q_{L'}$ is integral, likewise write $\bar{L}'=L'/\pi L'$. Integrality and zero-block conjecture. If $q_{M'}$ on $M'$ is integral, then $q_{L'}$ on $L'$ is integral, and hence these conditions are equivalent. In this situation, the dimensions of $\bar{L}'$ and $\bar{M}'$ modulo their radicals are equal. Equivalently, the number of zero entries in $\mathrm{GK}(L')$ equals the number of zero entries in $\mathrm{GK}(M')$. The conjecture would characterize the integrality of $q_{L'}$ through the summand $M'$ and relate the resulting reduced quadratic spaces to the zero entries of their Gross–Keating invariants; the paper proves it when $p$ is odd or when $L$ is anisotropic over $\mathbb{Z}_2$, while the general case remains open.