The uniform finite-order stability conjecture

Let $\lambda = \sup_{(m,a) \in M \times A}\int_A f(m,b)\operatorname{grad}_b f(m,a)\,db$ be finite, and suppose $\dim(M)=n$. For each $R>0$, consider domains $U$ with $\operatorname{diam}(U)<R$.

Uniform finite-order stability conjecture. There exists a constant $K(\lambda,n,R)$ such that stability for the local length-minimization problem in every such $U$ can be determined by examining $q$-parameter variations of order at most $q=K(\lambda,n,R)$.

This is presented as a more precise form of the finite-variation expectation. The source motivates it using bounded variability and compact domains, but provides no proof.