Pagani–Ricolfi–van Zelm's tautologicality conjecture for Brill–Noether pullbacks

Fix an integer $k$, integers $(a_1,\ldots,a_n)$, and a boundary divisor $D$ of the universal curve $\overline{\mathcal{C}}_{g,n}$. Let $A$ denote this data, and let

$$\mathsf{aj}_A:\overline{\mathcal{M}}_{g,n}\to\mathsf{Pic}_{g,n}^\theta$$

be the rational section sending $(C,x_1,\ldots,x_n)$ to $\omega^k(\sum a_i x_i)\otimes\mathcal{O}(D|_C)$. For the Brill–Noether class $\mathsf{w}_{g,d}^r(\theta)$, set

$$\mathsf{w}_{g,A,d}^r(\theta):=\mathsf{aj}_A^*(\mathsf{w}_{g,d}^r(\theta))\in\mathsf{CH}^{g-\rho}(\overline{\mathcal{M}}_{g,n}).$$

Pagani–Ricolfi–van Zelm's conjecture. The classes $\mathsf{w}_{g,A,d}^r(\theta)$ lie in the tautological ring $R^*(\overline{\mathcal{M}}_{g,n})$ for any choice of $g,d,A$ and generic $\theta$. This predicts that these pullbacks of Brill–Noether classes are tautological on the moduli space of stable pointed curves.