  The construction of creation operators of exact strings in eigenvectors of
the eight vertex model at elliptic roots of unity of the crossing parameter
which allow the generation of the complete set of degenerate eigenstates is
based on the conjecture that the 'naive' string operator vanishes. In this note
we present a proof of this conjecture. Furthermore we show that for chains of
odd length the string operator is either proportional to the symmetry operator
$S$ or vanishes depending on the precise form of the crossing parameter.
