Divisibility by powers of the new factors of Gram determinants

Let $G_n$ be the Gram matrix and let $H_n$ denote the factors of $\det G_n$ not occurring in $\det G_{n-1}$, so that

$$H_n\mid\det G_n,\qquad \gcd(H_n,\det G_{n-1})=0.$$

New-factor divisibility conjecture. For the resulting sequence of factors,

$$(H_{n-1})^{2n}\mid\det G_n.$$

The conjecture is motivated by observations for $\det G_1$ and $\det G_2$ and is part of the paper's speculative factorization program; no proof or resolution is supplied.