We investigate the problem of introducing consistent self-couplings in free theories
for mixed tensor gauge fields whose symmetry properties are characterized by
Young diagrams made of two columns of arbitrary (but different) lengths. We prove
that, in flat space, these theories admit no local, Poincaré-invariant, smooth, selfinteracting
deformation with at most two derivatives in the Lagrangian. Relaxing
the derivative and Lorentz-invariance assumptions, there still is no deformation that
modifies the gauge algebra, and in most cases no deformation that alters the gauge
transformations. Our approach is based on a Becchi-Rouet-Stora-Tyutin (BRST)
-cohomology deformation procedure