"....According to media reports, it was left to Turnbull to make...

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    "....According to media reports, it was left to Turnbull to make the decision...."

    Nothing in your post contradicts the above comment.



    Below from your link, you have Rudd's claim and a secondhand comment about accuracy from a spokesman, so hardly proof on your part by any stretch of the imagination.
    (Note also there was no comment about November by the spokesman)


    "....A spokesman for the prime minister said on Friday night Rudd’s representation of the December conversation was inaccurate.... “The claim there was support in December is wrong,” the prime minister’s spokesman said. “Rudd was told the decision would be made by the cabinet.”"

    “You in fact sent me a message on your preferred Wickr system where you stated that you and the FM [foreign minister] were “as one” in your support for my candidature.” Rudd said they met in November, then again in December.

    https://www.theguardian.com/austral...colm-turnbull-reneged-previous-support-un-job
 
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