Selberg-type second-moment conjecture for prime coefficients

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

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

for all $x\geq 2$. This is one of the additional assumptions used to obtain asymptotic estimates for $L$-functions in the Selberg class; the supplied text does not state whether it has been proved or remains open.