The finite constant-solution conjecture for deterministic state-space systems

Let $\,\Sc^{\mathrm{det}}$ denote the set of deterministic state–input solutions of a state-space system, let $\US^-$ be its input sequence space, and let $\#\Sc^{\mathrm{det}}(\Seq{u})$ denote the number of solutions associated with an input $\Seq{u}$. The system has the finite-memory property (FMP) when its solutions are determined by their input histories.

Finite constant-solution conjecture. If $\#\Sc^{\mathrm{det}}(\Seq{u})$ is constant on $\US^-$ and this constant is finite, then the state-space system has the FMP.

The conjecture proposes the converse of the result that the FMP implies a constant number of solutions. Its status is not specified in the source.