  We prove that a general complete intersection of dimension $n$, codimension
$c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if
$c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in
$N$ and quadratic in $n$. This degree bound substantially improves the
currently best-known super-exponential bound in $N$ by Deng, although our
result does not address the case $n \leq c < 2n-2$.
