Conjecture on unique solvability and guided cycles

Let $(\Gamma,\zeta,\Omega)$ be the guided dynamical system associated with the second partly characteristic boundary value problem, and let $\mathcal{N}^\Omega_\zeta$ denote the set of all $\Omega$-proper, $\Omega$-guided cycles in $(\Gamma,\zeta,\Omega)$. Then the boundary value problem

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Unique-solvability conjecture. The boundary value problem is uniquely solvable if and only if $\mathcal{N}^\Omega_\zeta=\emptyset$.

This conjecture seeks a simple necessary and sufficient condition for the guided dynamical system $(\Gamma,\zeta,\Omega)$, equivalently the associated $\mathcal{P}$-configuration, to be minimal. The preceding corollary establishes the analogous assertion under the assumptions of Proposition 2, but the stated general characterization is presented as a conjecture.