Note that we do not assume $\bar{\tau}\in\mathfrak{D}$. But by passing to the formal open subscheme of $\mathfrak{X}^{\prime}$ consisting of the formal open subsets $\mathfrak{U}$ with $S(\mathfrak{U})=\bar{\tau}$, we may assume $\Delta=\bar{\tau}$ and then the polytopal subdivision $\mathfrak{D}^{\prime}$ consisting the polytope $\bar{\tau}$ and its faces is suitable for $h$. The corresponding formal scheme is $\mathfrak{X}^{\prime}$. Let $D^{\prime}$ be the Cartier divisor on $\mathfrak{X}^{\prime}$ induced by $h$ as in Proposition 2.11. Notice that by construction we have $D=\iota^{\ast}D^{\prime}$. Now $\iota$ is proper by [Tem00, Corollary 4.4] (the result requires $\mathfrak{X}^{\prime\prime}$ to be admissible but by [Gro65, Proposition 2.7.1] it is enough to check properness after base change to the completion of an algebraic closure of $K$, after which $\mathfrak{X}^{\prime\prime}$ is always admissible, see Construction 2.6). Hence the projection formula yields $\Deg(D.Y)=\Deg(D^{\prime}.\iota_{\ast}Y)$. Now

$$\iota(Y)\subseteq\iota(\red_{\mathfrak{X}^{\prime\prime}}(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau)))=\red_{\mathfrak{X}^{\prime}}(p_{\mathfrak{X}^{\prime}}^{-1}(\tau)),$$

where the latter is the stratum in $\tilde{\mathfrak{X}}^{\prime}$ corresponding to $\tau$ and hence a point. Therefore $D^{\prime}.\iota_{\ast}Y=0$. ∎