The generating-function conjecture for motivic Omega-classes

For each positive integer $d$, define

$$\Omega_d(t)=\sum_{n\geq 0}\Omega^{n+2}_dt^n\in K_0(\operatorname{Var}_{\mathbb C})\llbracket t\rrbracket.$$

Let $\zeta_{{\mathbb P}^{d-1}}(t)$ denote the zeta function of projective $(d-1)$-space. Omega-class generating-function conjecture. Fix $d\geq 1$. If $d=1,2,3$, then

$$\Omega_d(t)=\zeta_{{\mathbb P}^{d-1}}(t)\cdot{\mathbb L}^{d-1}.$$

If $d>3$, then

$$\Omega_d(t)=\zeta_{{\mathbb P}^{d-1}}(t)\cdot\mathsf Q_d(t),$$

where $\mathsf Q_d(t)\in K_0(\operatorname{Var}_{\mathbb C})[t]$ is

$$\mathsf Q_d(t)=\sum_{i=0}^{d-3}\left(\sum_{j=0}^{i}(-1)^j\Omega^{i-j+2}_d[\operatorname{Gr}(j,d)]{\mathbb L}^{\binom j2}\right)t^i.$$

This is proposed as the analogue for the $\Omega$-classes of the paper’s main motivic formula.