Existence of an $L_{0}$-local Hamiltonian for $L_{0}$-regular spin chains

Let an $L_{0}$-regular spin chain be a spin chain with the regularity property specified in the source, and call a Hamiltonian $L_{0}$-local when it has interactions only between neighbours separated by at most $L_{0}-1$ sites.

Existence conjecture. For any $L_{0}$-regular spin chain there always exists a Hamiltonian which is $L_{0}$-local.

This asserts that the regularity condition is sufficient to construct a Hamiltonian with interaction range bounded by $L_{0}-1$. The supplied text gives no resolution status or further evidence, so the conjecture remains open.