Curzon is therefore writing (since this is what is being used in equation (5)) that any virtual, quasistatic process away from the final equilibrium state (to other equilibrium states on the constrained manifold describing the composite system) must obey $dS_i\geq 0$.
Curzon is not referring to quasistatic process when writing $dS_i\geq 0$. The relevant passage reads:
Consideration will now be given to the case where the barrier is perfect thermal insulator, all other properties of he barrier remaining the same as before. In this circumstance the barrier will only transmit mechanical energy; that is, $dU_1$ and $dV_1$will be independent. An equation to represent this interdependence may be obtained from the principle of increase of entropy, according to which the entropy of a thermally isolated system can only remain the same or increase. Subsystems 1 and 2 are each thermally isolated because by assumption the piston is made of a perfect thermal insulator. Therefore, from the principle of increase of entropy
$$
dS_1\geq 0,\tag{10}
$$
and
$$
dS_2\geq 0.\tag{11}
$$
Equations (5), (10), and (11) can only be consistent if
$$
dS_1 = dS_2 = 0.\tag{12}
$$
The problem discussed here by Curzon is the equilibrium state of an adiabatic system partitioned into two parts via a fixed adiabatic partition. Initially each compartment contains a gas at different temperature and pressure. The partition is then made movable and the question is to determine the pressure at equilibrium by application of the second law. The argument then is that during this process each compartment is a closed adiabatic system, therefore the second law applies individually to each compartment. At the final state each compartment must be in equilibrium, i.e., $dS_i=0$ with respect to variations about the equilibrium state. The maximization condition does not imply that $dS_i=0$ along the entire path, only at the state of equilibrium.
The purpose of the paper is to refute prior statements that the condition for mechanical equilibrium is $P_1/T_1=P_2/T_2$. Curzon argues that the correct condition is $P_1=P_2$ regardless of whether $T_1=T_2$ or not. This of course is obvious from a mechanical standpoint but Curzon's point is that the result is demanded by the second law.