Finite monoid catenary degree for conductor submonoids

Let $F$ be a factorial monoid and let $H\subset F$ be a conductor submonoid. The reduced monoid $F_{\mathrm{red}}$ is obtained by factoring out the units of $F$.

Finite catenary-degree conjecture. If $F_{\mathrm{red}}$ is finitely generated, then

$$\mathsf c_{\mathrm{mon}}(H)<\infty.$$

The conjecture asks whether finite generation of the reduced factorial ambient monoid forces the monoid catenary degree of every conductor submonoid to be finite. The surrounding results establish restrictions on adjacent catenary degrees and construct examples with arbitrarily large equal-length catenary degree, but do not resolve this finiteness question.