The recently established agreement between experiment and theory for the $g$ factors of lithiumlike silicon and calcium ions manifests the most stringent test of the many-electron bound-state quantum electrodynamics (QED) effects in the presence of a magnetic field. In this Letter, we present a significant simultaneous improvement of both theoretical $g_text{th} = 2.000,889,894,4,(34)$ and experimental $g_text{exp} = 2.000,889,888,45,(14)$ values of the $g$ factor of lithiumlike silicon $^{28}$Si$^{11+}$. The theoretical precision now is limited by the many-electron two-loop contributions of the bound-state QED. The experimental value is accurate enough to test these contributions on a few percent level.
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