  We determine spins of more than 100 low lying glueball states in $D=2+1$
dimensional $SU(4)$ gluodynamics by a lattice calculation. We go up to $J=8$ in
the spin value. We compare the resulting spectrum with predictions of the
Axionic String Ansatz (ASA). We find a perfect match for 39 lightest states,
corresponding to the first four string levels. In particular, this resolves
tensions between the ASA predictions and earlier spin determinations. The
observed spins of heavier glueballs are also in a good agreement with the ASA.
We did not identify any sharp tension between lattice data and the ASA, but
more work is needed to fully test the ASA predictions for the spins of 64
states at the fifth string level.
