A tellingly high proportion of the people polled were unsure — 42%, which was a strong second place, just behind the 46.5% who thought the chief executive should resign in such a scenario.
Just 11.5% thought the situation would call for the mayor to resign.
The poll of more than 1000 people seems to have particular relevance for Gore, where district council chief executive Stephen Parry and Mayor Ben Bell have not been on speaking terms.
The New Zealand Taxpayers’ Union commissioned Curia Market Research to ask people a question on the subject of "dysfunctional councils".
It did so between May 31 and June 6.
The question posed: If there is a breakdown in the relationship between a council mayor and chief executive that cannot otherwise be resolved, who should resign, the mayor or the chief executive?
Men were more certain of their view than women, and so were people aged 60-plus, compared with the younger demographics.
Generally, the strong consensus of the sure was against the chief executive.
The proportion of people aged 18 to 39 who thought the mayor should resign was relatively high, compared with the older demographics, at 22%.
"Unsure" was a clear winner among people who were also unsure about which political party they favoured.
A petition calling for Mr Parry to resign that was signed by more than 4800 people across New Zealand was this month presented to the Gore District Council.
The council decided not to accept it.
Mr Bell (24) has himself faced calls to resign.
Councillors had been poised to press ahead with a symbolic vote of no-confidence in the mayor last month, but backed out of it amid a strong public backlash.
Taxpayers’ Union campaigns manager Callum Purves said there was a power imbalance within councils.
"While voters rightly expect that the people they elect to represent them will be able to make a difference, this often isn’t the case," he said.
"The real power rests in the hands of the unelected officials who can sometimes see elected members as a problem to be dealt with, rather than the people calling the shots.
"We have seen this problem come into sharp focus with the problems in Gore where a longstanding council chief executive seems to be unwilling to work with their new democratically elected mayor.
"This poll result makes clear what New Zealanders think should happen in a situation like this.
"If someone needs to resign to allow the council to function, it must be the unelected bureaucrat, not the elected mayor."