The uniqueness and minimal-solution representation conjecture

Let $U$ be a set, let $f$ be known completely on $U$, and prescribe the boundary values $\psi|_{\partial U}$. Let $\psi$ satisfy the two equations denoted in the source by the relations for the model's length-minimization problem, and let $L$ be the associated length function.

Uniqueness and representation conjecture. The field $\psi$ is uniquely determined in $U$ by $f$, its boundary values, and those two equations. Moreover, $\psi$ can be expressed in terms of the minimal solutions of $L$ on $U$.

The source immediately qualifies this as potentially incorrect and separately poses the problem of computing all locally minimizing solutions when uniqueness fails. The conjecture is not resolved there.