The killer, described in the past as being "evil", has lost subsequent appeals that he was locked up unlawfully.
Now, the Court of Appeal has rejected his latest attempt.
Genge, now 43, had applied for orders setting aside the notice of abandonment and granting him leave to pursue the appeals earlier abandoned.
He also sought leave to bring appeals out of time in relation to the murder conviction and rape sentence.
Genge claimed that his decision to withdraw the appeals were affected by a wrongful process employed at the time by the Court of Appeal in dealing with criminal appeals reliant on legal aid - a process later described by the Privy Council in R v Taito as "a fundamentally flawed and unlawful system".
But in a decision just released, the Court of Appeal has rejected his case.
"The delay in bringing the appeal is extreme, Mr Genge has no explanation for it, and he has not identified a viable ground of appeal," the judgment concludes.
Genge remains a serving prisoner. He is eligible for parole but has been declined in the past.