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The council is due to consider the SHA at its meeting in Wanaka next Thursday.
If the council gives its approval, Associate Minister for Housing and Urban Development Jenny Salesa told the Otago Daily Times this week she "may consider" visiting Lake Hawea.
Residents who oppose the SHA are keen to discuss their concerns with the minister.
Ms Salesa is the minister responsible for SHAs and would receive a recommendation from the council, which she would then take to Cabinet for a final decision.
Ms Salesa said she could "only accept or reject a SHA proposal on the basis of whether it meets statutory requirements, as set out in the Special Housing Areas Act 2013".
"These requirements include demonstrating that the area will be serviced by infrastructure and that there is ongoing demand to build."
According to the Act, the minister is under "no obligation" to approve an SHA "even if the minister is satisfied that all criteria for making a recommendation are met".
Those opposing the SHA, which is intended to deliver affordable housing, believe the town already has sufficient land for that purpose.