The weighted derivative-substitution conjecture

Let $\alpha_1,\dots,\alpha_e$ be positive and let $n$ be a positive integer. Define a linear map on monomials by

$$\mathbf{x}^{\mathbf{I}}y_1^{t_1}\cdots y_e^{t_e}\longmapsto \mathbf{x}^{\mathbf{I}}(\alpha_1t_1+\cdots+\alpha_et_e)^n.$$

The weighted derivative-substitution conjecture. This mapping determines a map

$$\mathbf{P}^{\mathrm{pos}}_d\longrightarrow\mathbf{P}^{\mathrm{pos}}_{d+e}.$$

The source introduces this conjecture as arising naturally from the preceding composition arguments; its general validity is not established there.