There has been an interesting development in interpreting the conjecture in AdS [5, 6, 7, 8].

While the Cauchy-Schwarz inequality implies that correlators of a single interface operator and any number of bulk operators are bounded, it does not imply that the limit exists. Furthermore, the norm of multiple defect insertions is not automatically bounded. It might be possible to improve this by considering multiple interfaces, but we do not pursue this avenue here.

If the interface in the limit is not trivial but topological and noninvertible, defect operators may become operators with a topological line attached.

For arbitrary values of the coupling constant, Liouville theory has a symmetry-neutral $(1/2,1/2)$ operator, hence violating our assumption 3. This operator may not correspond to a normalizable state in Liouville theory, but our current argument is insensitive to such a normalizability issue.

From modular $T$ invariance, $$h-\overline{h}-\frac{c-\overline{c}}{24}\in \mathbb{Z},$$while the single-valuedness of correlation functions implies the quantization of spin, $h-\overline{h}\in \mathbb{Z}/2$. Combining these relations, we conclude that $c-\overline{c}$ must be quantized. This, in turn, means that $c-\overline{c}$ is invariant under continuous deformations.

Reflection positivity is sufficient only to show that (2) is bounded in the limit [16], while the smoothness condition implies the even stronger bound; ${|c_1-c_2|}^{2/3}/N_D$ stays finite in the limit $N_D\to 0$—see Supplemental Material [25].

Of course, if a scalar has dimension 1 only on some submanifold of the conformal manifold, our analysis works by considering points in the complement of said submanifold.