Ms Lockett believed Lundy was self-centred and would exhaust every possible legal outlet without considering other people's feelings.
She said it was a now a case of waiting with "bated breath once again" to see how the latest appeal panned out.
Ms Kincade said the appeal would be lodged in the Court of Appeal.
"The appeal is against sentence and conviction. It is not appropriate to rehearse in any detail in a public forum the grounds of the appeal, but it can be said that the issues raised will include a consideration of the [messenger RNA] evidence." Ribonucleic acid (RNA) was discussed during the most recent trial.
A key plank of the Crown case was that Mrs Lundy's brain matter was found in one of her husband's polo shirts.
Prosecutor Philip Morgan, QC, said testing showed two stains found on the shirt were from tissue one billion billion times more likely to belong to Mrs Lundy than any other person in New Zealand.
Dr Laetitia Sijen from the Netherlands Forensic Institute said she arrived at a conclusion brain matter was present on Lundy's shirt based on tests performed on RNA in a sample. But professor of molecular medicine Stephen Buston claimed there were problems with timings and the test.
Lundy was first found guilty of the murders in 2002 and sentenced to life. In 2013, Lundy appealed to the Privy Council, which threw out his conviction and a retrial began this year.