Divisibility by a binomial power of the first Gram determinant

Let $G_n$ be the Gram matrix in the paper, and let divisibility be taken in its polynomial ring. The first-determinant divisibility conjecture. For every integer $n\geq 1$,

$$(\det G_1)^{\binom{2n}{n-1}}\mid \det G_n.$$

The conjecture is motivated by computed low-dimensional cases and is offered as part of the paper's proposed factorization pattern; the source gives no proof or subsequent resolution.