Suppose there is an isolated system composed of two subsystems $A$ and $B$ in thermal contact with each other, but mechanically and diffusively insulated from each other.
The system starts off with a finite temperature difference between $A$ and $B$ (let $A$ have the higher temperature). $A$ then transfers heat to $B$. Can this transfer be quasistatic? If it was to be quasistatic, then $-\delta Q > T_A dS_A$ and $\delta Q > T_B dS_B $ would have to be strict. Then the discrepancies would need to be accounted for by assigning non-zero values to $\delta W$, $dV_A$, and/or $dV_B$ such that the equations $$-\delta Q -\delta W = T_A dS_A -P_A dV_A$$ and $$\delta Q +\delta W = T_B dS_B -P_B dV_B$$ hold. Is this possible to do?