The finite-variation stability conjecture

Consider a solution of the local length-minimization problem on a compact domain. Stability is tested using variational derivatives of the length functional.

Finite-variation stability conjecture. To determine whether the solution is stable over a compact domain, it suffices to examine derivatives of only finitely many orders.

The source gives an informal argument based on a bound on the domain diameter and remarks that analytic solvability of the induced partial differential equations may be needed. No rigorous resolution is supplied.