Conjecture on monotone catenary degree for generalized arithmetic monoids with gcd one

Let $M$ be the generalized arithmetic monoid under consideration, with parameters $h$ and $d$, and let $c(M)$, $c_{mon}(M)$, and $c_{eq}(M)$ denote its regular, monotone, and equivalent catenary degrees. The gcd-one case conjecture. If

$$\gcd(h-1,d)=1,$$

then the following cases hold: if $h<d$, then $c(M)<c_{mon}(M)$; if $h\geq d$ and $c(M)<c_{eq}(M)$, then $c(M)<c_{mon}(M)$; and if $h\geq d$ and $c(M)=c_{eq}(M)$, then $c(M)=c_{mon}(M)$. These conjectures aim to characterize the monotone catenary degree according to the gcd and the equivalent catenary degree; the source provides no resolution, so this assertion remains open.