Reflective Borcherds-product conjecture for free unitary modular-form algebras

Let $d$ be the discriminant parameter of the imaginary quadratic field underlying the unitary group, let $L$ be a Hermitian lattice of signature $(n,1)$, and let $\mathop{\mathrm{U}}(L)$ denote its unitary group. A finite-index subgroup is a subgroup of finite index in $\mathop{\mathrm{U}}(L)$. Reflective unitary freeness conjecture.

(i) When $d=-1$ or $d=-3$, the algebra of modular forms for a finite-index subgroup of some $\mathop{\mathrm{U}}(L)$ is never free if $n>5$.

(ii) When $|d|>3$, the algebra of modular forms for a finite-index subgroup of some $\mathop{\mathrm{U}}(L)$ is never free.

This is proposed as a unitary analogue of the result of Vinberg and Shvartsman that orthogonal modular-form algebras are never free when $n>10$. It arises from the expectation that the Jacobian of generators of a free unitary modular-form algebra should come from a reflective orthogonal Borcherds product by restriction; the conjecture remains open in the stated generality.