  The quark-hadron transition that happens in ultra-relativistic heavy-ion
collisions is expected to be influenced by the effects of rotation and magnetic
field, both present due to the geometry of a generic non-head-on impact. We
augment the conventional $T$--$\mu_B$ planar phase diagram for QCD matter by
extending it to a multi-dimensional domain spanned by temperature $T$, baryon
chemical potential $\mu_B$, external magnetic field $B$ and angular velocity
$\omega$. Using two independent approaches, one from a rapid rise in entropy
density and another dealing with a dip in the squared speed of sound, we
identify deconfinement in the framework of a modified statistical hadronization
model. We find that the deconfinement temperature $T_C(\mu_B,~\omega,~eB)$
decreases nearly monotonically with increasing $\mu_B,~\omega $ and $eB$ with
the most prominent drop (by nearly $40$ to $50$ MeV) in $T_C$ occurring when
all the three quasi-control (via collision energy and centrality) parameters
are simultaneously tuned to finite values that are typically achievable in
present and upcoming heavy-ion colliders.
