Conjecture on adjacent factorizations in generalized arithmetic monoids with gcd one

Let $M$ be the generalized arithmetic monoid under consideration, let $m\in M$, and let $\mathcal Z(m)$ be its set of factorizations. For factorizations $z,z'\in\mathcal Z(m)$, write $|z|$ for their lengths and $d(z,z')$ for their distance. Let $a$ be the parameter appearing in the source formulas. The adjacent-factorization conjecture. Assume

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

and that there exist $z,z'\in\mathcal Z(m)$ such that $z\ne z'$, $|z|=|z'|$, and $z_1>z'_1$. Then there exists $f\in\mathcal Z(m)$ such that

$$|f|=|z|+1,\qquad d(f,z)\leq d(z,z'),\qquad d(f,z')\leq d(z,z').$$

Further, if $h\leq d$, then

$$f=\left(z_1+\left\lceil\frac a2\right\rceil,z_2-\frac{a-\frac{ah+d-\left\lceil\frac a2\right\rceil(ah+d-a)}d}{ah+d-a},z_3-\frac{ah+d-\left\lceil\frac a2\right\rceil(ah+d-a)}d\right),$$

and if $h>d$, then

$$f=\left(z_1+\left\lfloor\frac a2\right\rfloor,z_2-\frac{a-\frac{ah+d-\left\lfloor\frac a2\right\rfloor(ah+d-a)}d}{ah+d-a},A z_3-\frac{ah+d-\left\lfloor\frac a2\right\rfloor(ah+d-a)}d\right).$$

The claim is intended to establish the length-adjacent factorization condition needed for strict inequality between regular and monotone catenary degrees in the gcd-one case; the source provides no resolution, so it remains open.