The last concrete sighting received by intelligence officials was that she had been seen late last year in different places close to the Syrian Iraq border.
There were hopes she would be found either when the IS stronghold of Baghouz was overrun, or picked up among the people trying to flee the area.
The International Committee of the Red Cross broke its silence in April, calling for information that could help locate Ms Akavi and the two drivers captured alongside her.
In April, Foreign Minister Winston Peters said the Ministry of Foreign Affairs and Trade had led a "significant and sustained whole-of-government effort ... involving a wide range of other agencies, and cooperation with international partners".
That included the deployment of a "small multi-agency team" based in Iraq, which was "specifically focused on locating Louisa and identifying opportunities to recover her".
The government is making no comment about the current status of that team, which included special forces troops from the New Zealand Defence Force.
Akavi was held captive alongside Kayla Mueller, a US aid worker, for what's thought to be at least four-and-a-half months.
The US military operation at the weekend against Abu Bakr al-Baghdadi was named after Mueller, in her honour.
Mueller is said to have been killed in February 2015; her mother Marsha told RNZ in April that Akavi had helped her daughter - and she hoped she would still be found alive.
"Kayla spent a lot of time in isolation while she was there - she was captured in August of 2013, you know, Louisa and her colleagues were captured in October.
"Kayla was with other women that we do know of, but for shorter periods of time and the information we got from them on how strong Kayla was, and how together they were able to cope so much better is amazing."
One possible theory had been that Akavi was kept close to Abu Bakr al-Baghdadi because of her nursing background, and his death raised hopes some information may have been unearthed after the raid.
However the government source said there was no evidence so far that she had been with him, or in the vicinity, but the government was still seeking any further information that may be relevant to her.
- RNZ