FPT status of length-constrained Disjoint Paths parameterized by k

Determine whether length-constrained variants of the Disjoint Paths problem (for example, minimizing the total length of the paths) are fixed-parameter tractable when parameterized by the number k of terminal pairs.

Background

While the classical Disjoint Paths problem is FPT when parameterized by the number of terminal pairs k (Robertson–Seymour theory), adding length constraints substantially complicates the problem. The authors note that the FPT status of such length-constrained variants under parameter k remains unresolved.

They highlight this as a major open question in the literature and discuss how their results yield subexponential algorithms parameterized by the total length l, but do not settle FPT parameterization by k for the length-constrained versions.