Make a new rule “Timer” with the text:

For each day after October 1st where the Katamari was not pushed by a Prince other than the King, the Timer is reduced by one. The Timer starts at 5 and is tracked on the Katamari wikipage. If the Timer reaches 0, any Prince may roll a DICEX where X is the is the number of non-King Princes. Only the first correct roll counts for this rule. The Prince that is selected from the sidebar, alphabetically first being 1 etc. and skipping the King, has achieved victory.