Universality of the Galois action on the thrice-punctured projective line

Let $F$ be a number field, let $p$ be a prime, and let $G_F$ be its absolute Galois group. For every tangential base point $0_v$ supported at $0$, consider

$$\overline{\mathbb{Q}}_p[\pi_1^{\mathrm{pro-alg}}({\mathbb P}^1_{\overline{F}}\setminus\{0,1,\infty\},0_v)]^{G_F-\mathrm{fin}}.$$

Universality conjecture. Any irreducible $\overline{\mathbb{Q}}_p$-representation of $G_F$ that is almost everywhere unramified and de Rham at places above $p$ can be established as a subquotient of this space for every tangential base point $0_v$ supported at $0$. The claim is presented as the second conjecture in the paper’s reformulation of the Fontaine–Mazur conjecture; the supplied text gives no resolution status.