This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
As I already told you at the workshop, Taras Banakh and me solved this problem. But, surprisingly and converse to that I told at the workshop, the answer is affirmative. This is why is good to write a common paper: if one of the authors is trying to prove a conjecture and another is trying to refute it then one of them should be right. :-)