Counterexample to a problem of Geoghegan-West

Let $X$ be a Banach space and $\mathrm{GL}(X)$ its general linear group. Let $\|\cdot\|$ denote the operator norm and ``$w$'' the pointwise convergence topology on $\mathrm{GL}(X)$. Is the identity map $(\mathrm{GL}(X),\|\cdot\|)\rightarrow(\mathrm{GL}(X),w)$ a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space $\mathbb J$---the ``space of counterexamples in Banach spaces theory''.