  We apply the extended local hidden gauge formalism to study the meson-meson
interactions with the quark constituents $cc\bar c\bar c$, $cc\bar c\bar b/\bar
c\bar c cb$, $cc\bar b\bar b/\bar c\bar c bb$, $bb\bar c\bar b/\bar b\bar b
cb$, and $bb\bar b\bar b$, where the exchanged mesons can only be the
fully-heavy vector mesons $J/\psi$, $B_c^*$, and $\Upsilon$. We solve the
coupled-channel Bethe-Salpeter equation to derive two poles in the $bb\bar
c\bar b$ system and two poles in the $cc\bar c\bar b$ system (with four
charge-conjugated poles in the $\bar b\bar b cb$ and $\bar c\bar c cb$
systems): one pole existing in the $bb\bar c\bar b$ system corresponds to the
sub-threshold bound state when setting the cutoff momentum $\Lambda > 850$ MeV,
while it causes the threshold effect with $\Lambda < 850$ MeV; the other pole
existing in this system corresponds to the sub-threshold bound state with
$\Lambda > 1100$ MeV; the two poles existing in the $cc\bar c\bar b$ system
correspond to the sub-threshold bound states only when setting $\Lambda > 1550$
MeV and $\Lambda > 2650$ MeV, making them difficult to be identified as
deeply-bound hadronic molecules. We propose to investigate the two poles of the
$bb\bar c\bar b$ system in the $\mu^+\mu^-B_c^-$ channel at LHC.
