  LiZn$_2$Mo$_3$O$_8$ is an electrically insulating geometrically frustrated
antiferromagnet in which inorganic Mo$_3$O$_{13}$ clusters each behaves as a
single $S = 1/2$ unit, with the clusters arranged on a two-dimensional
triangular lattice. Prior results have shown that LiZn$_2$Mo$_3$O$_8$ does not
exhibit static magnetic order down to at least $T = 0.05\,K$, and instead
possesses a valence bond ground state. Here, we show that LiZn$_2$Mo$_3$O$_8$
can be hole doped by oxidation with $\mathrm{I}_2$ and subsequent removal of
$\mathrm{Zn}^{2+}$ cations to access the entire range of electron count, from
one to zero unpaired electrons per site on the triangular lattice. Contrary to
expectations, no metallic state is induced; instead, the primary effect is to
suppress the number of sites contributing to the condensed valence-bond state.
Further, diffraction and pair-distribution function analysis show no evidence
for local Jahn-Teller distortions or other deviations from the parent trigonal
symmetry as a function of doping or temperature. Taken together, the data and
density functional theory calculations indicate that removal of electrons from
the magnetic layers favors Anderson localization of the resulting hole and an
increase in the electrical band-gap over the formation of a metallic and
superconducting state. These results put strong constraints on the chemical
conditions necessary to realize metallic states from parent insulating
geometrically frustrated antiferromagnets.
