Business case out but key details redacted
Harriet from Greater Auckland finally managed to get the Third Main Business Case from Kiwi Rail sent to her (via an Official Information Act request) although the report is still heavily redacted.
You can read the report below:
Some key points

The redacted material in that slide is pretty much outlined in this presentation slide from Kiwi Rail:

Source: Transport Blog via Kiwi Rail
I am taking a guess here but the Benefit Cost Ratio for the Third Main would be about 5.0 although using calculations that were used on the Holiday Highway you get an estimated BCR of 10.0. That is $500m worth of benefits for a $53 million project.
Compared to the East West Link (Option F):

Source: Greater Auckland
Furthermore, on the basis of more recent project estimates of $1.25-1.85 billion, the BCRs for the EWC would be 1.42 (on the $1.25 billion number) or 0.96 (on the $1.85 billion number).
“A BCR below 1.0 means that from an economic perspective, the country would be better off if the EWC were not built. While it is likely that other options considered at the IBC stage would also have encountered some of the same pressures (particularly in respect of escalation in construction cost), these numbers suggest that the original decision to prefer option F was not as well-grounded as it might have been. Options with higher BCRs, rejected by the NZTA, would have left significantly more headroom to absorb cost pressures and still result in projects with BCRs that made them worthwhile for society to construct.”
….
Source: https://www.greaterauckland.org.nz/2017/06/13/economics-east-west-link/
So the East West Link has an estimated BCR of less than 1.0 thus it should not be built but it is. While the Third Main has a BCR of anywhere between 5.0 and 10.0 and Finance Minister Steven Joyce denied it in Budget 2017. For Transport Minister Simon Bridges troubles the Barbara Streisand effect has now applied because if he was trying to hide the significant benefits of the Third Main pretty much everyone is now curious and wanting to know.
In the end the entire affair reeks of politics. We have a Government going full steam on marginal roading projects with no economic case behind them while rail projects that do (Third Main, and the City Rail Link) struggle to get into the Budget.

Source: Kiwi Rail