Prime-coefficient first-moment conjecture for the Selberg class

Let $\mathcal{L}\in\mathcal{S}$, and let $a(p)$ denote its prime coefficients. Let $\widehat{\mathcal{C}_{\mathcal{L}}^{P_1}}(x)$ and $\widehat{\mathcal{C}_{\mathcal{L}}^{P_2}}$ satisfy the stated growth and monotonicity conditions. Prime-coefficient first-moment conjecture. One has

$$\sum_{p\leq x}\left|a(p)\right|\leq \widehat{\mathcal{C}_{\mathcal{L}}^{P_1}}(x)\frac{x}{\log{x}} + \widehat{\mathcal{C}_{\mathcal{L}}^{P_2}}\frac{x}{\log^{2}{x}}$$

for all $x\geq 2$. This is presented as an additional assumption in the study of asymptotic estimates for Selberg-class $L$-functions; the supplied text gives no evidence resolving it.