We consider the higher-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, propagating 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, yielding an equation known to imply Fronsdal’s equation in the compensator form. We review the general context and results obtained in the investigation of metric-like higher-spin geometry, the structure of the corresponding non-local actions, together with their links to more conventional, local forms including a recently proposed one for higher-spin theories with transverse gauge invariance.