  A first-order differential equation is provided for a one-form, spin-s
connection valued in the two-row, width-(s-1) Young tableau of GL(5). The
connection is glued to a zero-form identified with the spin-s Cotton tensor.
The usual zero-Cotton equation for a symmetric, conformal spin-s tensor gauge
field in 3D is the flatness condition for the sum of the GL(5) spin-s and
background connections. This presentation of the equations allows to
reformulate in a compact way the cohomological problem studied in 1511.07389,
featuring the spin-s Schouten tensor. We provide full computational details for
spin 3 and 4 and present the general spin-s case in a compact way.
