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A$\newcommand\C{\Bbb C}\newcommand\om{\omega}$A counterexample is given by $E=\mathbb C^2$ and$K_i(x_1,x_2)=x_i$ for$i=1,2$$K_1(x_1,x_2)=(x_1,0)$ and$K_2(x_1,x_2)=(0,x_2)$ for complex $x_1,x_2$.
Indeed, then $\om(K_1)=\om(K_2)=1=\|(K_1,K_2)\|$.
A counterexample is given by $E=\mathbb C^2$ and$K_i(x_1,x_2)=x_i$ for$i=1,2$ and complex $x_1,x_2$.
$\newcommand\C{\Bbb C}\newcommand\om{\omega}$A counterexample is given by $E=\mathbb C^2$ and $K_1(x_1,x_2)=(x_1,0)$ and$K_2(x_1,x_2)=(0,x_2)$ for complex $x_1,x_2$.
