Robert Jennings, 51, and Scott Snoden, 39, were jailed for life after being found guilty of a fatal fire attack that claimed the lives of Thomas Sharkey Senior, 55, his son Thomas Junior, 21 and daughter Bridget, 8.
Both have launched appeals.
Jennings' counsel Gary Allan QC told the Court of Criminal Appeal in Edinburgh that there was not sufficient evidence to place Jennings as the perpetrator.
Mr Allan said: "Whilst there may be a basis for suspicion in this case it does not amount to more than suspicion."
The appeal judges will give a decision at a later date.