The exact motivic formula conjecture for the cusp singularity

Let $(X,p)$ be the cusp singularity $x^2=y^3$. For each $d\geq 0$, let $\mathit{NQ}_{d,X,p}^\mathrm{mot}(t)$ denote the normalized local motivic quotient series, and let $(a;q)_n$ and $(a;q)_\infty$ denote the finite and infinite $q$-Pochhammer symbols. Define polynomials $c_j(q)$ by

$$\sum_{j=0}^d c_j(q)t^j=(-t;q)_d.$$

The exact motivic formula conjecture.

$$\mathit{NQ}_{d,X,p}^\mathrm{mot}(t)=\sum_{j=0}^d \mathbb{L}^{\binom{j+1}{2}+j(d-j)}c_j(\mathbb{L})t^{2j},$$

and

$$\widehat{Z}_{X,p}^\mathrm{mot}(t)=\frac{1}{(\mathbb{L}^{-1}t;\mathbb{L}^{-1})_\infty}\sum_{n=0}^\infty\frac{\mathbb{L}^{-n^2}t^{2n}}{(\mathbb{L}^{-1};\mathbb{L}^{-1})_n}.$$

These formulas would give the first explicit motivic zeta-series formula for the cusp and imply the cusp case of the Cohen–Lenstra conjecture. The displayed identities are conjectural, despite consistency checks through $d\leq 30$.