Existence of a non-uniform graded ideal zeta function

For each $c\in\mathbb{N}$, let $\mathfrak{f}_{c,2}$ be the free class-$c$ nilpotent Lie ring on two generators, and let $\zeta_{\mathfrak{f}_{c,2}}^{\triangleleft_{\mathrm{gr}}}(s)$ denote its graded ideal zeta function over finite fields. Non-uniformity existence conjecture. There exists $c\in\mathbb{N}$ such that $\zeta_{\mathfrak{f}_{c,2}}^{\triangleleft_{\mathrm{gr}}}(s)$ is $\mathbb{F}_p$-non-uniform. The case $c=6$ is already known not to be $\mathbb{F}_p$-uniform, but the source notes that the stronger distinction between finite uniformity and non-uniformity is not yet known there.