  We study the consequences of a hadron-quark phase transition (PT) in failing
core-collapse supernovae (CCSNe) which give birth to stellar-mass black holes
(BH). We perform a suite of neutrino-transport general-relativistic
hydrodynamic simulations in spherical symmetry with 21 progenitor models and a
hybrid equation of state (EoS) including hadrons and quarks. We find that the
effect of the PT on the CCSN postbounce dynamics is a function of the bounce
compactness parameter $\xi_{2.2}$. For $\xi_{2.2}\gtrsim0.24$, the PT leads to
a second dynamical collapse of the protocompact star (PCS). While BH formation
starts immediately after this second collapse for models with
$\xi_{2.2}\gtrsim0.51$, the PCS experiences a second bounce and oscillations
for models with $0.24\lesssim\xi_{2.2}\lesssim0.51$. These models emit potent
oscillatory neutrino signals with a period of $\sim$ms for tens of ms after the
second bounce, which can be a strong indicator of the PT in failing CCSNe if
detected in the future. However, no shock revival occurs and BH formation
inevitably takes place in our spherically-symmetric simulations. Furthermore,
via a diagram of mass-specific entropy evolution of the PCS, the progenitor
dependence can be understood through the appearance of third-family of compact
stars emerging at large entropy induced by the PT.
