A phase transition for $q$-TASEP with a few slower particles

We consider a $q$-TASEP model started from step initial condition where all but finitely many particles have speed $1$ and a few particles are slower. It is shown in [9] that the rescaled particles position of $q$-TASEP with identical hopping rates obeys a central limit theorem `a la Tracy-Widom. We adapt this work to the case of different hopping rates and show that one observes the so-called BBP transition. Our proof is a refinement of Ferrari-Vet"o’s and does not require any condition on the parameter $q$ nor the macroscopic position of particles.