I agree with everyone who has said that he must go to court if he is to have any chance. If the ex-wife lies, she lies. If she succeeds in convincing the judge that her lies are true, then he can beg the court to allow him supervised visits with an intermediary. In fact, because of the woman's history of lying, I would seriously consider the following course of action:

Take it to court and request shared custody or visitation. If he is granted shared custody, and you can afford it, set up a surveillance system in your home and record during the times that she is there.

If he is granted visitation, hire a third party to supervise visitation without bringing it to the court's or her mother's attention.

In both of these instances, it is likely that she will lie to the court about what has been happening to her daughter while with your husband and it is likely (based on other such cases) that she will lie sooner rather than later. In this way, there will be an impartial record of his behavior. If you can prove that the mother is lying about abuse or other inappropriate behavior via an impartial method, then she'll be found guilty of perjury, and he'll quickly find the court on his side of future custody matters.

If he is denied both of these requests, he can beg the court for supervised visits. They stink, but if he makes every visit for an extended amount of time, he'll be able to petition the court for unsupervised visits in the future - at which point he should then continue to make certain the visits are supervised to prevent sabotage through perjury.

If all of the above attempts fail, I would recommend that he blog his love and heartache for his daughter beginning immediately. Five years from now, ten, fifteen...but she will find it some day - and it will be a record of his love and yearning for his daughter that will pay dividends if they must begin their relationship anew later in life.