The foliation $\mathcal{F}$ can be, in fact, continued to the boundary of ${\Delta_{\nu}}$ (not smoothly at the $d-2$-skeleton of ${\Delta_{\nu}}$) via the diffeomorphism $\mu\circ\mathrm{Log}_{s}^{-1}$ between $\mathbb{R}^{d}$ and the interior of ${\Delta_{\nu}}$. So that it will induce a projection $X_{\Delta_{\nu}}\backslash\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{\{0\}}(\epsilon))\to\Sigma$. But to construct torus fibrations we will use only a part of this projection where it is clearly well defined. That is why we do not provide a proof for this more general statement here.