The Lieb theorem is a cornerstone of quantum magnetism theory in condensed matter. In this work, we investigate the instability of the Lieb insulating ferrimagnetic phase in the extended Hubbard model on a trimer chain at half-filling, with one electron per site, under increasing the nearest-neighbor Coulomb coupling $V$. Our results show that despite a noticeable increase in doublon density with $V$, the ferrimagnetic insulating phase remains robust up to the phase separation (PS) line, which is observed at $V \gtrsim U/4$, where $U$ is the local Coulomb repulsion. Above the PS line, one of the coexisting phases is primarily populated by doublons on one of the two sublattices of the chain. This phase coexists with a metallic, unsaturated ferromagnetic phase for $U \gtrsim t$, and with a singlet phase for $U \lesssim t$, where $t$ is the intra-trimer hopping amplitude. We estimate the PS and the crossover lines with the help of density matrix renormalization group calculations.

10 pages, 10 figures