Newton polytope equality for theta basis and other pointed bases

Prove that for any degree m in the lattice M associated to a seed of a cluster algebra, the m-pointed theta basis element θ_m and the corresponding m-pointed element S_m from any M-pointed basis S (for example, the common triangular basis L or the generic basis) have the same Newton polytope in the ambient torus algebra LP. Establishing this equality would extend valuative results known for the theta basis to other pointed bases.

Background

The paper discusses valuative characterizations of partially compactified cluster algebras via tropical points and notes recent progress proving such results for the theta basis. Extending these valuative properties to other natural bases, such as the common triangular basis or the generic basis, would require a structural comparison of their Newton polytopes with those of theta functions.

The authors formulate a conjecture that equality of Newton polytopes for basis elements would imply analogous valuative conclusions, thereby enabling parallel compactification and optimization results for non-theta bases.