Linear independence of time-frequency translates for ultimately positive functions

We prove that the HRT conjecture holds when the Gabor system consists of a 4-point set in the time-frequency plane and a square-integrable function that is ultimately positive. We also prove the conjecture for Gabor systems generated by an ultimately positive function and translation sets, whose frequencies satisfy at most one linear dependence over $\mathbb{Z}$, improving a result of Benedetto and Bourouihiya.