You may like to check with your 2nd storey neighbors whether they are facing the same problem. Most probably they do not because the partial choke is usually located just below the floor of the affected unit. That is, the choke in the other waste pipe should be near the roof of the 2nd storey, usually at the interconnecting pipe that links the 2 waste pipes. The choke could be caused by rag getting caught at the interconnecting pipe joint. This can happen when there is renovation being carried out in one of the upper units and the worker carelessly drop rags, sponge into the waste pipe.
I am not sure if TC can / will "rod" the other waste pipe from the manhole, all the way to the 3rd storey. Usually, if the ground floor is void deck, TC can "rod" to the 3rd storey by opening the inspection "eye" of the other waste pipe from the void deck.
Hope the TC contractor has the necessary experience to solve the problem.