  We consider the most general, classically-conformal, three-dimensional
$\mathcal{N}=1$ Chern-Simons-matter theory with global symmetry $Sp(2)$ and
gauge group $U(N)\times U(N)$. We show that the Lagrangian in the on-shell
formulation of the theory admits one more free parameter as compared to the
theory formulated in off-shell $\mathcal{N}=1$ superspace. The theory on $T^3$
can be formally localized. We partially carry out the localization procedure
for the theory on $T^3$ with periodic boundary conditions. In particular we
show that restricting to the saddle points with vanishing gauge connection
gives a trivial contribution to the partition function, i.e. the bosonic and
fermionic contributions exactly cancel each other.
