Generally speaking, this does not seem like a problem: The guy sent a photograph of two clothed people and a bunch of text describing sexual activity between consenting adults. Some ways that it could be a problem, which disagree with the facts presented:
- True depictions/descriptions of sexual activity with a minor
- True depictions/descriptions of non-consensual sexual activity with anybody
- Distributing pornographic/sexual images of another person without that person's permission (depending on jurisdiction)
- Falsely claiming as true untrue statements about another person that would tend to lower that person's reputation in the community (defamation)
- If the man had/wanted a security clearance to access government secrets, and the conversation content revealed material that could be used for blackmail OR demonstrated extremely poor judgment/handling of confidential information
- If the transcript was shared with and believed to be true by someone the man wanted to be partnered with, the probability of that partnership might be reduced (not a legal problem, but still potentially viewed as a problem).
In the first two categories, if the descriptions were made/interpreted, regardless of truth, they could provide some reason to investigate further to see if they were true or not. If an investigator concludes they are probably not true, there shouldn't be an issue. If an investigator believes that the depictions/descriptions in combination with other evidence prove beyond a reasonable doubt that a crime was truly committed, they could prosecute that. (The legal standard to start prosecution is lower, but they like a pretty high win percentage and to focus resources where they have more to go on.)
If investigators get a copy of the chat and the man were a suspect in some related crime, they might try to use it to show state of mind (e.g. "the woman was killed/raped/etc, and here's this chat from our prime suspect her ex showing how upset and emotionally agitated he was two hours before.") The conversation itself, as described, does not seem to be a crime.
Also, context matters. People in private Facebook chats with friends have no legal obligation to tell the truth, and sometimes lie. People, especially men, in private conversation sometimes invent fictions with no claim to the truth and/or brag about sexual activities regardless of whether or not those statements are true. (As the US president might say, "that's just locker room talk.") Were there to be any more serious charges of legal issues, the alleged offender would be given a chance to explain himself and frame the conversation in context appropriately and attempt to dispel misinterpretations. However, the people who investigate such allegations are human and able to recognize context too; the conversation described seems unlikely to draw any sustained attention.