  The bound-electron g factor is a stringent tool for tests of the Standard
Model and the search for new physics. The comparison between an experiment on
the g factor of lithiumlike silicon and the two recent theoretical values
revealed the discrepancies of $1.7\sigma$ [D. A. Glazov $\textit{et al}$.,
Phys. Rev. Lett. $\textbf{123}$, 173001 (2019)] and $5.2\sigma$ [V. A. Yerokhin
$\textit{et al}$., Phys. Rev. A $\textbf{102}$, 022815 (2020)]. To identify the
reason for this disagreement, we accomplish large-scale high-precision
computation of the interelectronic-interaction and many-electron QED
corrections. The calculations are performed within the extended Furry picture
of QED, and the dependence of the final values on the choice of the binding
potential is carefully analyzed. As a result, we significantly improve the
agreement between the theory and experiment for the g factor of lithiumlike
silicon. We also report the most accurate theoretical prediction to date for
lithiumlike calcium, which perfectly agrees with the experimental value.
