The weight decomposition conjecture for $L_{10}$

Let $L_{10}=P_{10,425}$ and let $\mathcal{W}_{12}$ denote the weight-12 component. $L_{10}$ weight decomposition conjecture. The period $L_{10}$ mixes weights $12$ through $14$, and

$$\mathcal{W}_{12}L_{10}\in\mathbb{Q}Q_{12,4}+\mathbb{Q}^{\times}Q_{12,5}+\mathcal{W}_{12}\mathcal{P}_{\phi^4,\leq8}.$$

The statement is presented as following from Scenario 1 and the preceding conjectures, but the displayed decomposition is not established in the supplied text; it remains conjectural.