Extension of pseudo-ASL term orders to the polynomial ring

Determine whether every pseudo-ASL term order on a finitely generated commutative algebra A extends to a term order on the ambient polynomial ring R[H] compatible with the straightening structure.

Background

In graded ASLs some relative term orders on the polynomial ring are known, but in general the relationship between pseudo-ASL term orders on A and term orders on R[H] is unclear. Establishing such an extension would connect intrinsic and ambient Gröbner frameworks.