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Robin Dicey, Annabel Blaikie, Werner Murray and Shirley Calvert have requisitioned a September 19 board meeting to remove Mr Gillespie as board chairman, but have not said what is behind their move.
Fellow board members Nigel McKinlay and Anna Harrison have praised Mr Gillespie's chairmanship and said they were surprised by the move, and Central Otago Mayor Tim Cadogan announced his disappointment about the group's bid on his mayoral Facebook page on Tuesday.
It has since received considerable public comment, both on that page and after being shared on the "You know you are from Cromwell when" Facebook page.
A handful of people have supported the group's move, comments including "change can be a great thing for a community", "a bold move team" and that Mr Gillespie was part of a group with an "old school management style".
But most of the dozens of Facebook comments have wondered about the reasons for the move and/or supported Mr Gillespie.
One person said they were ''surprised and perplexed by this move by the so-called 'Gang of Four', another said the issue had ''arrived out of left field with very little information available'' and another said it was ''a sad outlook for Cromwell''.
Former Cromwell board member Helen Hucklebridge said she ''suspected'' the board ''had major problems, but this latest move proves that the board is completely in the control of four members''.
''During my 15 years as a CCB member and deputy chair for 12 of those years I found that chairman Neil's knowledge, his understanding of required procedures and his impartiality were all first rate.
''However I guess it all boils down to numbers and 'The Gang of 4' have the strength to impose their ideas on our community''.
Cromwell businessman Gary Anderson said it was "very disappointing to see this action being taken by the 4 named members, first the [Cromwell] memorial hall now this, what next if some people don't get their own way".
Fellow Cromwell businessman Greg Wilkinson, while also posting informative comments explaining how the community board operated, likened the incident to the line "Something is rotten in the state of Denmark" from Shakespeare's Hamlet.
He said it "could equally apply to Cromwell during the term of this current CCB . . . it seems the current CCB has lurched from one battle to the next, tackling a variety of issues which have drawn battle-lines around the CCB table and across our community".
Mr Cadogan said he "shared the same disappointment and confusion" as the "vast majority" of those who had made Facebook comments about the attempted ousting of Mr Gillespie.
He again criticised the group of four for not making its reasons for the move known.
An initial intention to requisition a meeting from the group said Mr Gillespie "has many responsibilities and commitments to the district" and, as well as being CCB chairman, "also needs to take a role of making district wide decisions, where there is a requirement to make decisions from the perspective of the wider district as a whole".
It also said "we feel that it is in the best interests of the CCB" that the chairmanship pass to someone who has "more time and effort" for the Cromwell Eye to the Future Masterplan process.
But the group's subsequent formal requisition for a meeting did not mention any reasons behind the move.
Mr Cadogan said it also needed to be noted that Mr Dicey and Mr Werner had both said at an informal CCB discussion they felt Mr Gillespie "was doing a good job as chair, but that it was their preference that a non-councillor be the chairperson moving forward".
However, Ms Calvert, who "has made it plain to me that she believes she's ready to take up the role of [CCB] chairperson", was also a Cromwell councillor as well as a CCB member, Mr Cadogan said.
A statement by Ms Calvert on Tuesday to the Otago Daily Times said: "In Neil we have an experienced chair who can be a helpful mentor to the board. So asking for this resolution is a procedural matter. Standing orders for the Cromwell Community Board require us to follow this process".
She would not comment further, and Mr Cadogan said Ms Calvert had refused to meet with Mr Gillespie to discuss any concerns.
The four members who had requisitioned the meeting had also "either ignored or declined" an offer to provide a trained, external facilitator to discuss any issues, and "have been either unwilling or unable to advise who they anticipate replacing Neil Gillespie with should the motion be successful", Mr Cadogan said.