  We consider high-derivative equations obtained setting to zero the divergence
of the higher-spin curvatures in metric-like form, showing their equivalence to
the second-order equations emerging from the tensionless limit of open string
field theory, which propagate reducible spectra of particles with different
spins. This result can be viewed as complementary to the possibility of setting
to zero a single trace of the higher-spin field strengths, which yields an
equation known to imply Fronsdal's equation in the compensator form. Higher
traces and divergences of the curvatures produce a whole pattern of
high-derivative equations whose systematics is also presented.
