The Fermat-Weber problem is one of the most widely studied problems in classical location theory. In his previous work, Brimberg (1995) attempts to resolve a conjecture posed by Chandrasekaran and Tamir (1989) on a convergence property of the Weiszfeld algorithm, a well-known iterative procedure used to solve this problem. More recently, Canovas, Marin and Caflavate (2002) provide counterexamples that appear to reopen the question. However, they do not attempt to reconcile their counterexamples with the previous work. We now show that in the light of these counterexamples, the proof is readily modified and the conjecture of Chandrasekaran and Tamir reclosed.