Strict monotonicity of the equilibrium parameters in non-symmetric Showcase Showdown

For each integer $n>1$, let $(\epsilon_n,\delta_n)$ be the unique solution in $[0,1]\times[0,1]$ of

$$\left\{ \begin{aligned} \mathrm{a)}\;(1+e^{x}(-1+x))^{n-2}&= \dfrac{e^{y}\,\left( -1+{\left( 1+e^{x}\,\left( -1+y\right) \right) }^{n}\right) +n\,e^{x}}{n\,e^{x}\, \left( 1+e^{y }\,\left( -1+y\right) \right) \,\left( 1+e^{x}\,\left( -2+n+x\right) \right) },\\ \mathrm{b)}\;{(1+e^{x}(-1+x))}^{n-1}&=\frac{e^{-x} \left(n e^x \left(e^x (y-1)+1\right)^{n-1}+\left(e^x (y-1)+1\right)^n-1\right)}{n y}. \end{aligned} \right.$$

with $\delta_n>\epsilon_n$. Strict-monotonicity conjecture. The sequences $\{\epsilon_n\}_{n>1}$ and $\{\delta_n\}_{n>1}$ are strictly increasing. The conjecture concerns the equilibrium parameters of the no-information, non-symmetric, constant-sum version of Showcase Showdown; the paper reports that the claim is suggested by computations for $2\leq n\leq 6$ but does not provide a proof for every $n>1$.