  In contrast to other atomic systems, in true muonium ($\mu^+\mu^-$) the
leading-order $Z$ boson corrections to the hyperfine splitting are shown to be
experimentally accessible in the near future. This contribution ($-109$ MHz)
constitutes a necessary contribution to a full $\mathcal{O}(m\alpha^7)$
calculation of the true muonium hyperfine splitting. This would enable a number
of solutions to the muon problem to be constrained. Additionally, we compute
the general expression for a pseudovector coupling to particle-antiparticle
bound states at leading order, including the annihilation channel.
