The modified log Manin conjecture for geometrically $\mathbb A^1$-connected varieties

Let $(\underline{X},\Delta)$ be a log pair and set $\underline{U}=\underline{X}\setminus\operatorname{Supp}(\Delta)$. Assume that $\underline{U}$ is geometrically $\mathbb A^1$-connected. Modified log Manin conjecture. Santens's log Manin conjecture for integral points holds under the additional assumption that $\underline{U}$ is geometrically $\mathbb A^1$-connected. The conjecture modifies Santens's formulation by replacing the assumption $\overline{F}[\underline{U}]^\times=\overline{F}^\times$ with geometric $\mathbb A^1$-connectedness; the paper proposes this condition because the preceding example shows that trivial geometric units do not generally imply $\mathbb A^1$-connectedness.