The problem of interest is to schedule two types of jobs on a two-stage flowshop with a common second-stage machine so as to minimize the makespan. This problem was previously known to be NP-hard in the ordinary sense. In this note, we give a proof for the strong NP-hardness. Scope and purpose Two different products are to be processed on two different machines but they must be then processed by a common machine, say for final testing. Given two sets of jobs, the goal is to find a production schedule achieving the shortest completion time. In this note, we establish the inherent intractability of the problem.