Is there an effect of a nontrivial $c_T$ during inflation?

Recently, we have shown that the propagation speed $c_T$ of the primordial gravitational waves (GWs) might be nontrivially varying during inflation, which could induce local oscillations in the power spectrum of primordial GWs. In this paper, we numerically confirm that, although with a disformal redefinition of the metric the nontrivial $c_T$ may be set as unity, the power spectrum in the frame with $c_T=1$ is completely the same as that in the original disformal frame, i.e., the oscillating shape in the power spectrum is still reserved, since here the effect of $c_T$ is actually encoded in the nontrivially varying Hubble parameter. In addition, we also clarify how to obtain a blue-tilted GWs spectrum by imposing a rapidly decreasing $c_T$ during inflation.