Lapid--Mao conjecture on adjoint L-values and Whittaker periods

Let $\Pi$ be a globally generic representation satisfying the paper's newvector assumption, and let $f=\bigotimes_p f_p\in\Pi$ be a newform. Let $W$ and $W_\infty$ be the Whittaker functions associated with $f$ and $f_\infty$, respectively. Lapid--Mao conjecture.

$$\frac{\langle f,f\rangle}{|W(1)|^2}\sim_{\mathbb{Q}^{\times}}\frac{L(1,\Pi,\operatorname{Ad})}{\Delta_G(1)|W_\infty(1)|^2}.$$

Here $\langle f,f\rangle$ is the $L^2$-norm, and $\Delta_G(s)$ is the $L$-function of the dual Artin motive attached to $G$. The paper invokes this conjecture as an ingredient in evidence for its period conjecture; it is not proved in the supplied text.