Ribbon-partition conjecture for q-Catalan germs

Let $\mathcal{L}=\mathbb{Z}^{a-1}$ be the weight lattice, let $\operatorname{Box}[c'+1,c]$ be the indicated slab of the fundamental box, and order lattice points by the tilted partial order. A ribbon is a saturated chain of length $a$ in this order. Ribbon-partition conjecture. If $1\leq c\leq(a-1)^2$, $\gcd(a,c)=1$, and $c'$ is the largest integer less than $c$ satisfying $\gcd(a,c')=1$, then the tilted partial order on

$$\mathcal{L}\cap\operatorname{Box}[c'+1,c]$$

has a partition into disjoint ribbons. The conjecture strengthens the standard-partition conjecture and is supported only by small cases $a\in\{3,4,5\}$, so it remains open.