Realize any submonoid of amenable functions as Px

Prove that for every submonoid A of the monoid Am (the set of amenable functions f : [0, ∞) → [0, ∞) under composition, where amenable means f−1(0) = {0}), there exists a subclass X of the class M of metric spaces such that Px = A; here Px denotes the set of all functions f : [0, ∞) → [0, ∞) satisfying (X, d) ∈ X implies (X, f ∘ d) ∈ X for every metric space (X, d).

Background

The paper studies, for a given class X of metric spaces, the monoid Px of functions f : [0, ∞) → [0, ∞) that preserve X under composition with metrics (i.e., (X, d) ∈ X ⇒ (X, f ∘ d) ∈ X). It characterizes when the equation Px = A has a solution X for various choices of A, including complete solutions for A = F, F0, and Am, and necessary and sufficient conditions when A ⊆ PM or A ⊆ PU.

Theorem 23 shows that Px = Am is achievable for certain discrete classes X, while Theorem 29 provides a solvability criterion Px = A for A ⊆ PM. Conjecture 32 extends these results by proposing that any submonoid of Am can be realized exactly as Px for some class X, beyond the constraints of PM.