The power-structure compatibility conjecture for the -Euler characteristic

Let $k$ be a field of characteristic not $2$, let $X/k$ be a quasi-projective variety, and let $n\geq 1$. Write $\operatorname{Sym}^n(X)=X^n/S_n$ for the $n$th symmetric power, let $\chi_c$ denote the $\mathbb{A}^1$-Euler characteristic, and let $a_*$ be the power structure on $\operatorname{GW}(k)$ constructed by Pajwani and Pál.

Power-structure compatibility conjecture. For every such $k$, $X$, and $n$,

$$\chi_c(\operatorname{Sym}^n(X))=a_n(\chi_c(X)).$$

Equivalently, $\chi_c$ respects the power structures when $\widetilde{K_0}(\operatorname{Var}_k)$ is equipped with the symmetric-power power structure and $\operatorname{GW}(k)$ with $a_*$. The conjecture asks for a quadratic refinement of the formula for the Euler characteristic of symmetric powers and was also formulated by Bejleri and McKean. It is currently unknown whether the stated compatibility holds.