= Solution
Preserve the primary <constraints> using the corrected <Hamiltonian> from <4/c/solution>. After a spatial integration by parts, the $B_0$ terms in its density are $B_0\partial_i\pi_i+B_0^*\partial_i\pi_i^*-M^2B_0^*B_0$. <Hamilton's equations> give
$$
0=\dot\pi_0=-\partial_i\pi_i+M^2B_0^*,\qquad0=\dot\pi_0^*=-\partial_i\pi_i^*+M^2B_0.
$$
With the definitions of $E^i,E^{*i}$ in the paper, the actual massive <Gauss law> is therefore
$$
\boxed{\partial_iE^{*i}=M^2B_0^*,\qquad\partial_iE^i=M^2B_0.}
$$
This is also the $\nu=0$ component of the <Proca equation>. The zero right-hand sides printed in the source apply only to the massless theory or to special configurations with $B_0=0$.
For a concrete counterexample, take $k\ne0$, $\omega^2=k^2+M^2$, and the longitudinal on-shell <plane wave>
$$
B^\mu=\left(\frac kM,0,0,\frac\omega M\right)e^{-i\omega t+ikz}.
$$
It satisfies $\partial_\mu B^\mu=0$, but $E^3=-iM e^{-i\omega t+ikz}$ and $\partial_iE^i=kM e^{-i\omega t+ikz}\ne0$. Imposing the printed zero-divergence condition would incorrectly remove the physical longitudinal <particle polarization>.
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