Polynomial-time Hermite normal form conjecture for pseudo-matrices over $A_{fi}$

Let $A_{fi}$ be the function-field ring considered in the paper, and let a pseudo-matrix over $A_{fi}$ represent a module by its coefficient ideals and matrix. Polynomial-time Hermite normal form conjecture. There exists a polynomial-time algorithm to compute the Hermite normal form of a pseudo-matrix over $A_{fi}$. The analogous polynomial-time result is known for maximal orders in number fields, but the corresponding result for maximal orders in function fields is not known in the literature; the paper leaves its investigation to further work.