The exterior-determination conjecture for nonunique solutions

Let $U$, $f$, the boundary conditions, and the equations be as in the preceding uniqueness conjecture. Suppose additionally that $\psi$ is known precisely in a neighborhood of $U$ contained in $M\setminus U^{\mathrm{int}}$.

Exterior-determination conjecture. The field $\psi$ is then determined precisely inside $U$.

This is proposed as a further claim in the event that the preceding interior problem has nonunique solutions. The source gives no argument establishing it.