Thermodynamic equilibrium - will the piston move?

Friend asked me this question and I didn't manage to solve it with basic thermodynamic reasoning. I believe this problem is easly solvable via numeric methods by choosing specific systems, though I prefer an analytic, more general and more intuitive solution.

Two different and isolated systems (which specified by $S_1(E_1,V_1,N_1)$ and $S_2(E_2,V_2,N_2)$) are seperatly prepeard to satisfy particular $(P,T)$ requirements, so that $P_1=P_2=P$ but $T_1 \ne T_2$. Afterwards the two systems are brough one near the other, with a single piston (unmovable at first) seperating them. The piston doesn't allow transfer of heat or particles at any stage. After the two systems were properly juxtaposed the restriction on the movement of the piston is removed. Will the piston move from its original position?

One way of treatment suggests that since $P_1=P_2$ and and since only mechanichal work (exchange) is allowed - the piston will not move.

Other way sugest that by forcing maximum entropy (thermodynamic equilibrium) for the combined system, we will get $dS_{tot}=dS_1+dS_2=0$, and in particular (since there is only one degree of freedom here) $\frac{\partial S_1}{\partial V_1}=\frac{\partial S_2}{\partial V_1}$ so at equilibrium $\frac{P_1}{T_1}=\frac{P_2}{T_2}$, hence the piston will move.