  On the basis of comparing eigenvalues of an operator ${\hat {\mathbf{\cal
C}}}{}^{({\cal R})}$, that proved useful in distinguishing how off-shell 4D,
$\cal N$ = 1 supermultiplets become off-shell 4D, $\cal N$ = 2 supermultiplets,
the double tensor supermultiplet is shown to be radically different for other
known multiplets. This suggests difficulties, if not impossibilities, to embed
it into an off-shell structure.
