Heat continues to flow between the tubs through the conductors, from hotter areas to colder ones, until the temperature difference between them is (including the connectors) zero, i.e. they are all at the same temperature $T_e$ (Zeroth Law of thermodynamics): full thermodynamic equilibrium has been reached and there's no more heat flow.
Because the system is adiabatic ($\Sigma Q=0$), we can write (initial state v. end state):
$$m_1cT_1+m_2cT_2+m_3cT_3=m_1cT_e+m_2cT_e+m_3cT_e$$
Assuming $m_1c=m_2c=m_3c$ ($m$ is water mass, $c$ water specific heat),
we get simply:
$$T_e=\frac{T_1+T_2+T_3}{3}$$
The nature of the connectors does not affect the end result, only the time needed to reach full thermodynamic equilibrium.