No answer, just a comment to illustrate the functionality of the Schröder-function which is not familiar for several participiants here.
I applied the Schröder-mechanism which runs into a power series involving complex terms. Here is a graph for iterates in steps of 1/60 from a couple of starting points in the interval $0 \ldots \pi/2$ Near that two borders, the Schröder-function is difficult to handle and the iterates in that neighbourhoods are questionable.
For starting points below $\pi/4$ :
For starting points above $\pi/4$ :
Of course, the spirals/trajectories can be continued infinitely towards the fixpoint.