  Finding a material that turns superconducting under ambient conditions has
been the goal of over a century of research, and recently
Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O aka LK-99 has been put forward as a possible
contestant. In this work, we study the possibility of electronically driven
superconductivity in LK-99 also allowing for electron or hole doping. We use an
$\textit{ab initio}$ derived two-band model of the Cu $e_g$ orbitals for which
we determine interaction values from the constrained random phase approximation
(cRPA). For this two-band model we perform calculations in the fluctuation
exchange (FLEX) approach to assess the strength of orbital and spin
fluctuations. We scan over a broad range of parameters and enforce no magnetic
or orbital symmetry breaking. Even under optimized conditions for
superconductivity, spin and orbital fluctuations turn out to be too weak for
superconductivity anywhere near to room-temperature. We contrast this finding
to non-self-consistent RPA, where it is possible to induce spin-singlet
$d$-wave superconductivity at $T_{\mathrm{c}}\geq300$ K if the system is put
close enough to a magnetic instability.
