Bejleri–McKean conjecture on symmetric powers of null motivic Euler characteristic

Let $k$ be a field of characteristic not $2$. Let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of $k$-varieties, let $\operatorname{GW}(k)$ be the Grothendieck–Witt ring, and let

$$\chi^c:K_0(\operatorname{Var}_k)\to\operatorname{GW}(k)$$

be the motivic, or compactly supported, Euler characteristic. For a quasi-projective $k$-variety $X$, write $[X]$ for its class and let $\operatorname{Sym}^n X$ denote its $n$th symmetric power. Bejleri–McKean's conjecture. If $X$ is a quasi-projective $k$-variety with $[X]\in\ker\chi^c$, then

$$[\operatorname{Sym}^n X]\in\ker\chi^c$$

for all $n\geq 1$; equivalently, the symmetric-power maps preserve $\ker\chi^c$. This conjecture would yield a power structure on $\operatorname{GW}(k)$ compatible with the motivic Euler characteristic and the power structure on the Grothendieck ring of varieties, and would support an enrichment of Göttsche's formula for Euler characteristics of Hilbert schemes.