  We prove a conjecture by Zhang, Ahmadain, and Klich that the spin-$1$ Motzkin
chain is gapped for any area weight $t<1$. Existence of a finite spectral gap
is necessary for the Motzkin Hamiltonian to belong to the Haldane phase, which
has been argued to potentially be the case in recent work of Barbiero,
Dell'Anna, Trombettoni, and Korepin. Our proof rests on the combinatorial
structure of the ground space and the analytical verification of a finite-size
criterion.
