Conjecture on solutions of the Hirzebruch functional equation and the associated differential equation

Let $f(z)$ be a solution of the Hirzebruch functional equation with $n>2$ and initial conditions

$$f(0)=0,$$ $$f'(0)=1.$$

Let the associated equation be the differential equation denoted by in the source.

Solution conjecture. Every solution $f(z)$ of the Hirzebruch functional equation with $n>2$ and these initial conditions is a solution of the associated differential equation.

The paper reports evidence for this claim in small values of $n$. It would identify all normalized solutions of the Hirzebruch functional equation in the range $n>2$ with solutions of the differential equation, but the general assertion remains open.