  We investigate whether Swampland constraints on the low-energy dynamics of
weakly coupled, moduli-stabilised string vacua in AdS can be related to
inconsistencies of their putative holographic duals or, more generally, recast
in terms of CFT data. We find that various swampland consistency constraints
are equivalent to a negativity condition on the sign of certain mixed anomalous
dimensions. This condition is similar to well-established CFT positivity bounds
arising from causality and unitarity, but not known to hold in general. The
studied scenarios include LVS, KKLT, and both perturbative and racetrack
stabilisation. Interestingly, the LVS vacuum (with $\Delta_{\varphi} = 8.038$)
also appears to live very close to a critical value ($\Delta_{\varphi} = 8$)
where the anomalous dimensions change sign. We finally point out an intriguing
connection to the Swampland Distance Conjecture, both in its original and
refined versions.
