Let $0< t \le 0.1$, $1<C<2$ and $0< x,y < 1/2$. Prove that
$$ (1 - e^{-\frac{x}{t}})(1 - e^{-\frac{y}{t}}) \ge C (e^{\frac{xy}{t}} -1)(e^{-\frac{x}{t}} + e^{-\frac{y}{t}}) $$
It is interesting to see that this question has two facets:
As $t$ is fixed and $x,y \to 0$, we can expand both sides to get an idea about the convergence behavior. We obtain $\frac{xy}{t^2} \ge 2 C t \frac{xy}{t^2}$ which is fine as $2 C t < 1$.
As $x,y$ are fixed and $t\to 0$, we write $$ (1 - e^{-\frac{x}{t}})(1 - e^{-\frac{y}{t}}) \ge C (e^{-\frac{x(1-y)}{t}} + e^{-\frac{y(1-x)}{t}}) - C(e^{-\frac{x}{t}} + e^{-\frac{y}{t}}) $$ and since $x,y < 1/2$, we get by expansion: $1 \ge 0$ which is fine.
The question is how to manage a curve discussion which brings the two ends into a common form. An approach (which I didn't manage to finish) is to multiply the question with $e^{\frac{x+y}{2t}}$ which gives
$$ 2 \sinh\frac{x}{2t}\sinh\frac{y}{2t}\ge C (e^{\frac{xy}{t}} -1)\cosh\frac{x-y}{2t} $$ and continue from here.
Credits go to user Hendrra for inspiring this question.