After conversion of an idea to a project, the methodology field is empty.
This is surprising as this field is the attribute 'project template' and the project is created using a template during the conversion.
Based on our testing and research it is expected that you make sure Project Template has a value, so for example if you add the attribute to the Project List View and only filter for Templates you will find a number of out of the box templates PMO once that you will notice have that value populated. Now you will also notice that some of the custom templates will not have that value populated so basically in order for when you convert an idea to project using a given template and if Project Template is not populated on the template itself it will not be shown on the project but if it is populated you should be able to see it when converting an idea using a project template. Please let us know if you have any further questions on this.