I expect that both the proliferations and the scape are photosynthesizing from the start. Unfortunately the question is whether they are synthesizing enough to balance the amount of any resources they are using that are supplied by the crown/original fan.
We might compare, at least theoretically, the behaviour of the plant with a scape that say has had all its buds eaten, a scape with only proliferations and a rhizome with a fan at its end. A scape that has had its buds eaten seems to be aborted relatively quickly. A scape with proliferations seems to last a relatively long time. A rhizome with a fan at its end presumably lasts a very long time. The fan on the end of the rhizome presumably makes little demand on the resources of the original crown and becomes independent relatively quickly. A scape with proliferations presumably makes less demand on the resources of the original crown. A scape that has lost its buds presumably does not supply enough resources to the crown to balance its potential dangers (easier disease/pest entry perhaps).
