Constructive automata conjecture for Ising-model languages

Let $\mathcal{M}$ be an Ising model and let ${\sf L}_{\mathcal{M}}$ be its language. A constructive PDA decides ${\sf L}_{\mathcal{M}}$ by accumulating the unary energy from local contributions and comparing it with the input energy; a constructive LBA accumulates the Hamiltonian energy on a designated tape in binary and accepts exactly when it agrees with the input.

Constructive automata conjecture. For every context free ${\sf L}_{\mathcal{M}}$ there exists a constructive PDA that decides it. For every context sensitive ${\sf L}_{\mathcal{M}}$ there exists a constructive LBA that decides it.

The claim says that the natural, function-computing condition imposed on the automata is not necessary for restricting the language classes: every language in the relevant class should admit such a constructive recognizer. The source gives no resolution of this conjecture.