fa.functional analysis - Generating cones having no surjections [in operator spaces]

Is this little toy known ?

Let $E$ be some Banach space, and let $K$ be the closed unit ball
of its dual, endowed with the weak-star topology. Also, let $j:E$ $rightarrow$ $C(K)$
be the natural embedding. Then, if $pi$ :$E$ $rightarrow$ $C(K)$
is onto, one must have $leftVert pi-jrightVert $ > 1. [Applying
this to $E=$ $ell^{1}$, or to $E=C[0,1]$ (eventually, via Milutin)
would be interesting, I think.]