Commutativity of QM transformation and discretization for QP systems

Let a QP differential system be given, and consider the multiplicative and additive discretizations

$$x_i(p+1)=x_i(p)\varphi_i\left(\lambda_i+\sum_{j=1}^{m}A_{ij}\prod_{k=1}^{n}x_k^{B_{jk}}(p)\right),\qquad i=1,\ldots,n,$$

के and

$$x_i(p+1)=x_i(p)+\varphi_i\left(\lambda_i+\sum_{j=1}^{m}A_{ij}\prod_{k=1}^{n}x_k^{B_{jk}}(p)\right),\qquad i=1,\ldots,n,$$

where each $\varphi_i$ is differentiable. Commutativity conjecture. For every QP differential system, the operations of QM transformation and discretization are commutative if and only if, in the multiplicative case, $\varphi_i(\xi)=a^\xi$ for all $i=1,\ldots,n$, with $a\in\mathbb{R}$; in the additive case, they are never commutative. The claim characterizes the QP discretization as the unique discretization of the multiplicative form with the desired commutativity property and rules out the additive form. The source presents this as a statement based on an examination of transformation properties, but the supplied text does not establish whether it is proved or remains conjectural.