The Modified Bellows Conjecture for flexible spherical polyhedra

Let $P(u)$ be a flexible polyhedron in the sphere $\mathbb{S}^n$, where $n\ge 3$. Replacing a vertex by its antipode means replacing that vertex of $P(u)$ with its antipodal point in $\mathbb{S}^n$. Modified Bellows Conjecture. Some vertices can be replaced by their antipodes so that the oriented volume of the resulting flexible polyhedron $P'(u)$ remains constant during the flexion. This conjecture is a spherical analogue of the Bellows Conjecture, which asserts constancy of oriented volume during flexion. It is refuted by exotic flexible cross-polytopes in $\mathbb{S}^n$ for all $n\ge 4$, while the source notes that it holds for the previously constructed counterexamples to the usual Bellows Conjecture.