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Under the hypotheses of analyticity in the coupling constant, locality,Lorentz covariance, and Poincare invariance of the deformations, combined withthe preservation of the number of derivatives on each field, the consistentinteractions between a collection of free massless tensor gauge fields with themixed symmetry of a two-column Young diagram of the type (3,1) and one Abelianvector field, respectively a $p$-form gauge field, are addressed. The mainresult is that a single mixed symmetry tensor field from the collection getscoupled to the vector field/$p$-form. Our final result resembles to the wellknown fact from General Relativity according to which there is one graviton ina given world.

No cross-interactions among different tensor fields with the mixed symmetry (3,1) intermediated by a vector field