Infinitesimal rigidity of hinged spanning cycles on the hypercube

Let the $n$-Roberts graph encode the facet adjacencies of the $n$-cube, and let a hinged spanning cycle be a spanning cycle of this graph whose corresponding $2n$ facet adjacencies are retained as hinges while all other ridges are removed. Infinitesimal rigidity conjecture. All spanning cycles of the $n$-Roberts graph, corresponding to hinged spanning cycles of facets on the $n$-cube, produce an infinitesimally rigid structure: there is no non-trivial first-order deformation that restricts to an infinitesimal congruence on each facet. This conjecture concerns the rigidity of prefabricated hypercube structures formed by hinging facets along the edges of a spanning cycle; establishing it would explain why such structures can be transported in a folded state before unfolding after an additional cut.