The European Parliament in a Resolution of 23 November 1989 invited "the parliaments of the Member States to the European Assizes, an assembly of the parliaments of Europe, to discuss the next stages of the implementation of the European Union". It instructed its Committee on Institutional Affairs to draw up a report on preparation for the Assizes. I understand that this report is still in preparation.
The reponsibility for the convening of the Assizes rests entirely with the European Parliament, in collaboration with the parliaments of the member states. The Presidency of the Council has no role or function in respect of these arrangements.