Heat continues to flow between the tubs through the conductors, until the temperature difference between them is $0$, i.e. they are all at the same temperature $T_e$ (Zeroth Law of thermodynamics): full thermodynamic equilibrium has been reached.
Because the system is adiabatic ($\Sigma Q=0$), we can write:
$$m_1cT_1+m_2cT_2+m_3cT_3=m_1cT_e+m_2cT_e+m_3cT_e$$
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.