Inconsistency of spin 4-spin 2 gauge field couplings

The authors invert the usual problem of coupling higher spin gauge fields to gravity by treating (linearised) gravity as the 'matter field' source of spin 4 gauge theory. This is motivated by the existence of the conserved gravitational four-index Bel-Robinson tensor as a possible current for the spin 4 field. They first derive this tensor as a Noether current, thereby linking it to a novel invariance of spin 2. It is then shown that, as usual for higher spins, consistency does not survive beyond lowest-order cubic coupling