A further meeting will be needed to decide the fate of Mr Tomás Ó Dulaing, the principal of Gaelscoil Thulach na nÓg in Dunboyne, Co Meath, who is in conflict with the school's board of management over religion policy.
A nine-hour hearing in Dublin on Saturday into the principal's case failed to reach a conclusion and a further meeting is to take place this week. The patron body for the school, An Foras Patrúnachta, has been asked by the school's board of management to dismiss Mr Ó Dulaing.
The INTO is strongly defending the principal and its officials have warned that dismissing him would be totally unacceptable. While some sources said that a certain level of "shuttle diplomacy" was taking place, there is little sign of a compromise emerging.
The row at the school, which has caused great bitterness among parents, concerns whether Catholic and Protestant children should be taught religion together at all times.
Mr Ó Dulaing has suggested that points of difference between the religions - such as Communion - should be removed from lessons during the day and taught instead outside school hours. However, this runs contrary to the views of the patron body and the board.
While this is the main cause of the dispute, personal differences between some of the main protagonists have also played a part.
Saturday's hearing was attended by the board, An Foras Patrúnachta, two INTO representatives (Ms Anne Kilduff and Mr Noel Ward) and Mr Ó Dulaing.
The hearing was "very polite", according to sources, although Mr Ó Dulaing spent much of it defending his decision of March 15th to send a letter to parents criticising the board's policy.
While the chance of a compromise still exists, the board of the school remains adamant that Mr Ó Dulaing must go. Foras is under pressure to accede to this wish, but it also has to take account of the deep opposition from the INTO, which has members in the 37 other schools presided over by Foras. The general secretary of Foras, Mr John Carr, attended the hearing at St Patrick's Training Centre in Drumcondra throughout Saturday.
The board of management will not be running the school in future, it was announced recently. Instead, Foras is to appoint a manager who will run the school for a time, with some assistance from the board members.
Meanwhile, Brother Luke Monahan, of the Marino Institute, is trying to act as facilitator between the two sides. However, his chances of success will probably depend on the outcome of the hearing.