Connectivity of the space of $l_p^n$s

Question

Is it known whether the space $\{ l_p^n : 1 < p < \infty\}$ is connected with the topology induced by Banach-Mazur distance? As stated in a comment, this would follow from a bound on the operator norm of the identity matrix mapping $l_p^n$ to $l_q^n$ in terms of $|p-q|$, so my question becomes: What is the operator norm of the identity matrix mapping $l_p^n$ to $l_q^n$?