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On the basis of comparing eigenvalues of an operator ${\hat {\mathbf{\calC}}}{}^{({\cal R})}$, 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 otherknown multiplets. This suggests difficulties, if not impossibilities, to embedit into an off-shell structure.

A Letter Highlighting Matrix Mapping in Minimal 4D, ${\mathbf {\cal N}}$ = 1 On-Shell Supermultiplet Representations