Converse to the finite-core-point criterion up to normalizer equivalence

Let $G\leqslant \operatorname{S}_{d}$ be a transitive permutation group, and let $\operatorname{Fix}(G)^{\perp}$ denote the orthogonal complement of the fixed space of $G$. A subspace is rational if it has a basis contained in $\mathbb{Q}^{d}$, and two core points are normalizer equivalent when they lie in the same orbit under the normalizer of $G$ in $\operatorname{GL}(d,\mathbb{Z})$.

Normalizer-equivalence conjecture. If $\operatorname{Fix}(G)^{\perp}$ contains a rational $G$-invariant subspace other than $\{0\}$ and $\operatorname{Fix}(G)^{\perp}$ itself, then there are infinitely many core points up to normalizer equivalence.

The conjecture is presented as the converse of the paper's finite-core-point criterion. The supplied text does not state whether it has been resolved, so its status is recorded as open.