Unconstrained local Lagrangians for higher-spin gauge theories are bound to involve auxiliary fields, whose integration in the partition function generates geometric, effective actions expressed in terms of curvatures. When applied to the triplets, emerging from the tensionless limit of open string field theory, the same procedure yields interesting alternative forms of geometric Lagrangians, expressible for both bosons and fermions as squares of field-strengths. This shows that higher-spin curvatures might play a role in the dynamics, regardless of whether the Fronsdal-Labastida constraints are assumed or forgone.