Kalton-Peck space is not isomorphic to its hyperplanes

Let $Z_2$ denote the real Kalton-Peck space. No hyperplane of $Z_2$ admits a complex structure, and $Z_2$ is even in the sense of Ferenczi-Galego. Therefore, $Z_2$ is not linearly isomorphic to its hyperplanes, solving a long-standing conjecture in Banach space theory. The argument uses the Kalton-Swanson symplectic form on $Z_2$, the Castillo-Gonz'alez-Pino Fredholm theorem in $Z_2$, and a special case of a mod-$2$ theorem of Li-Zhu on skew-adjoint Fredholm relations in symplectic spaces, for which a self-contained proof is included.