The problem with that Fox / Wilson effort, if it's the one I recall, is that it assumes its own conclusion. Let's assume it was JC, here's a bit of forensic geography, oo look, that proves it was JC.
There are a number of details and people I'd like to think were properly looked into at the time, or have been since. In no particular order:
1/ TS / PSS. Everything we think we know about these two came from PSS, none of it is corroborated especially the stuff about blind dates with Arab princes, PSS later accused TS of being scary, SJL was said to have told her sister (or was it a friend?) she was seeing a married man with kids; TS was a married man with kids, and one under financial stress - he went bankrupt 8 days later.
2/ The house next to SJL's was being done up into flats by a Cypriot property developer. The Greek for Cyprus is Kiprus. Kiprianou's a common name there. Mr who, again?