The hypergeometric description conjecture for classical-group de Rham local systems

Let $G$) be one of the classical groups under consideration, with standard representation $\mathrm{Std}$, and let $(\delta=0,\mu=\rho\times\phi)$ be rigid hypergeometric automorphic data. Write the parameters of $\rho$ as $\rho_1,\dots,\rho_{n-m}$, and let $\Hyp_\lambda$ denote the corresponding hypergeometric local system. Hypergeometric description conjecture. There exists a $\lambda\in\mathbb{C}^{\times}$, depending on $\phi$, such that

$$\cE^{\dR}_{\SO_{2n+1}}(0,\mu)(\Std)\simeq\Hyp_\lambda(\underline{0};\rho_1,\dots,\rho_{n-m},-\rho_1,\dots,-\rho_{n-m},1/2),$$ $$\cE^{\dR}_{\Sp_{2n}}(0,\mu)(\Std)\simeq\Hyp_\lambda(\underline{0};\rho_1,\dots,\rho_{n-m},-\rho_1,\dots,-\rho_{n-m}),$$

and, for $d=2m>n+1$,

$$\cE^{\dR}_{\SO_{2n+2}}(0,\mu)(\Std)\simeq\cE^{\dR}_{\SO_{2n+1}}(0,\mu)(\Std)\oplus(\mathscr{O}_{\mathbb{G}_m},d).$$

These formulas would give an explicit description of the de Rham local systems attached to standard representations of the classical groups and imply the expected functoriality of their Hecke eigenvalues under the relevant pushouts. The source does not provide evidence that the conjecture has been proved or disproved.