Let $w(x,y)=\frac{\log(\Lambda+\xi)-\log\left(\frac{1}{2}\Lambda-\frac{1}{2}\sqrt{\Lambda^2-4\frac{x-y}{\zeta}}\right)}{\log(\Lambda+\xi)-\log(\Lambda-\xi)}$ for $\Lambda>0$, $\xi>0$, $\xi\in\left(\frac{\Lambda}{2},\Lambda\right)$, $\zeta>0$.
Let $x^\star$ and $y^\star$ solve $$ x^\star = \max_x x w(x,y^\star)$$ and $$ y^\star = \max_y y \left(1-w(x^\star,y)\right)$$
If $\frac{\partial w}{\partial \Lambda}>0$ (which is equivalent with setting $x^\star>y^\star$), I need to show that $\frac{d w\left(x^\star, y^\star\right)}{d \Lambda}>0$.