Sign-change invariance conjecture for holonomy angles

Let $\Gamma$ be a lattice with trace field $K$, where $K$ is the field generated by the traces of elements of $\Gamma$. Let $h_K$ and $h_K^+$ denote the class number and narrow class number of $K$, respectively. A sign change is an element of $\{\pm1\}^n$ acting by changing the signs of the holonomy angles. Sign-change invariance conjecture. If

$$h_K=h_K^+,$$

then the holonomy angles of $\Gamma$ are invariant under all sign changes. This would extend the theorem for lattices derived from maximal orders in quaternion algebras over number fields with equal class and narrow class numbers, and for principal congruence groups in such lattices; the general case is left open by the source.