The composition conjecture

Let $f(\mathbf{x},y_1,\dots,y_e)\in\widehat{\mathbf{P}}_{d+e}$, and choose $g_i(y)\in\widehat{\mathbf{P}}^{\mathrm{pos}}$. Define

$$F(\mathbf{x},y)=f(\mathbf{x},g_1(y),\dots,g_e(y))=\sum f_i(\mathbf{x})y^i.$$

The composition conjecture. Every coefficient $f_i$ belongs to $\overline{\mathbf{P}}_d$, and

$$f_0\longleftarrow f_1\longleftarrow f_2\longleftarrow\cdots.$$

The paper investigates special cases, including one exponential function, but leaves the general composition assertion open.