  There exists a unique class of local Higher Spin Gravities with propagating
massless fields in $4d$ - Chiral Higher Spin Gravity. Originally, it was
formulated in the light-cone gauge. We construct a covariant form of this
theory as a Free Differential Algebra up to NLO, i.e. at the level of equations
of motion. It also contains the recently discovered covariant forms of the
higher spin extensions of SDYM and SDGR, as well as SDYM and SDGR themselves.
From the mathematical viewpoint the result is equivalent to taking the minimal
model (in the sense of $L_\infty$-algebras) of the jet-space extension of the
BV-BRST formulation of Chiral Higher Spin Gravity, thereby, containing also
information about (presymplectic AKSZ) action, counterterms, anomalies, etc.
