Asymptotic cut in half of a $d$-dimensional ball

Question

Denote by $\Delta_{d-1}=\{\mathbf{x}\in\mathbb{R}^d:x_1+x_2+\cdots+x_d=1\}$ the $d-1$ simplex and define $R=\{\mathbf{x}\in\mathbb{R}^d: d^3\|\mathbf{x}\|^2_2 - \|1/\mathbf{x}\|_1 \geq 0\}$ where $1/\mathbf{x} = (1/x_1, 1/x_2,...,1/x_d)$. Is it possible to show that $$\lim_{\varepsilon \to 0} \frac{\textrm{Vol}[\Delta_{d-1} \cap R\cap B_{\mathbb{R}^d}(\mathbf{1}/d, \varepsilon)]}{\textrm{Vol}[\Delta_{d-1} \cap R^c\cap B_{\mathbb{R}^d}(\mathbf{1}/d, \varepsilon)]} = 1 $$ i.e. that, restricted to the hyperplane $\Delta_{d-1}$, the region $R$ asymptotically splits the ball centered at $\mathbf{1}/d = (1/d, 1/d,..., 1/d)$ in two equal halves ?