Massive scalar particles cannot have local interactions with chiral higher-spin backgrounds, so no higher-spin extension of Riemannian geometry of this type exists.
Dirac,Forms of Relativistic Dynamics, Rev
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Inconsistency of point-particle dynamics on higher-spin backgrounds: massive particles
Massive scalar particles cannot have local interactions with chiral higher-spin backgrounds, so no higher-spin extension of Riemannian geometry of this type exists.