OnMouseDown is not invoked for the first click in WebPlayer

OnMouseEnter and OnMouseExit work fine. However, if player clicks, the first click does not make effect. The second click works. Actually, it does not mater where to click for the first time. It looks like the WebPlayer is not in focus. In the same time, working OnMouseEnter and OnMouseExit point to the fact that WebPlayer listen the mouse events in general. Debug.Log(Input.mousePosition) inside the Update() also prints out the current position.
Is it a way to detect the first click?