= Solution
The unintegrated closed-string operator $c\widetilde c\,\mathcal O$ must have total conformal weight $(0,0)$ and ghost number $(1,1)$. Since $c$ and $\widetilde c$ have weights $(-1,0)$ and $(0,-1)$, the matter operator $\mathcal O$ must be a <Virasoro primary operator> of weight
$$
(h,\widetilde h)=(1,1).
$$
Equivalently, the full operator must be a <BRST-closed operator> and not a <BRST-exact operator>. For a momentum-dependent tensor operator, these requirements impose its target-space mass-shell, transversality, and gauge-equivalence conditions.
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