Lifting Galois sections along torsors

The cuspidalization conjecture, which is a consequence of Grothendieck’s section conjecture, asserts that for any smooth hyperbolic curve $X$ over a finitely generated field $k$ of characteristic $0$ and any non empty Zariski open $U\subset X$, every section of ${\pi }_1(X,\bar{x})\to {\mathrm{G}\mathrm{a}\mathrm{l}}_k$ lifts to a section of ${\pi }_1(U,\bar{x})\to {\mathrm{G}\mathrm{a}\mathrm{l}}_k$. We consider in this article the problem of lifting Galois sections to the intermediate quotient ${\pi }_1^{cc}(U)$ introduced by Mochizuki. We show that when $k = \mathbb Q$and$D=X\setminus U$ is an union of torsion sub-packets every Galois section actually lifts to ${\pi }_1^{cc}(U)$. One of the main tools in the proof is the construction of torus torsors $F_D$ and $E_D$ over $X$ and the geometric interpretation ${\pi }_1^{cc}(U)\simeq {\pi }_1(F_D)$.