Standard-partition conjecture for q-Catalan germs

Let $\mathcal{L}=\mathbb{Z}^{a-1}$, let $\operatorname{Box}$ be the fundamental box, and for coprime $a,c$ let $c'$ be the largest integer less than $c$ with $\gcd(a,c')=1$. A standard set is a set of lattice points whose tilted-height generating function is a monomial times $[a]_q$. Standard-partition conjecture. For every integer $c$ with $1\leq c\leq(a-1)^2$ and $\gcd(a,c)=1$, the set

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

has a standard partition. Such partitions would support the positivity and lattice interpretation of q-Catalan germs; the source leaves the assertion as a conjecture.