Does there exist a positive constant $c>0$ such that for any two positive real numbers $0<\sigma_1\le\sigma_2$,
$$ (\sigma_1-1)^2+(\sigma_2-1)^2-(1-2\sigma_1 \sigma_2) \ge c \big( \sigma_1^2 +(\sigma_2-1)^2 \big)$$
holds?
I know that $c \le 1$ (take $\sigma_1 \to 0$), and that $c=1$ does not universally holds; In fact, the inequality holds for $c=1$ exactly when $2\sigma_2 \ge \sigma_1+2$. (since we assumed that $\sigma_1\le\sigma_2$ this happens whenever $\sigma_1 \ge 2$ for instance).
Taking $\sigma_2 \to \infty$ seems to also work for any $c>0$.
No, because if $\sigma_1 = \frac13$ and $\sigma_2 = \frac23$ then the left hand side is $0$ and the right hand side is $\frac{2c}9 > 0.$