Conundrum for the free energy of a holonomous gluonic plasma at cubic order

We compute the term $\sim g^3$ in the free energy for a $SU(N)$ gauge theory with nonzero holonomy at nonzero temperature. If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order $g^3$, ${\mathcal{F}}^{\left(3:\mathrm{c}\mathrm{.}\mathrm{g}\mathrm{.}\right)}$, is singular: ${\mathcal{F}}^{\left(3:\mathrm{c}\mathrm{.}\mathrm{g}\mathrm{.}\right)}\mathrm{\ne }0$ without holonomy, but ${\cal F}^{\left( 3 \rm{:c.g.} \right)}= 0$ when the holonomy is nonzero, even infinitesimally. We show that the absence of the charged gluon contribution is required by gauge invariance alone and is therefore a universal feature.