Does the a-anomaly decrease along all six-dimensional RG flows
In plain words
In two and four dimensions a number counting effective degrees of freedom is proven to drop under coarse-graining. In six dimensions this is proven only for special supersymmetric cases.
Precise statement
In unitary 6D QFT, the Euler coefficient a of the trace anomaly is conjectured to satisfy $a_{UV} > a_{IR}$ for every RG flow between CFTs. The 4D dilaton proof (Komargodski and Schwimmer 2011) does not carry over because the relevant six-derivative dilaton coupling is not sign-definite by unitarity. Prove or refute the inequality for general, including non-supersymmetric, flows.
What would settle it
A general argument (dilaton, entanglement or other) establishing $a_{\mathrm{UV}} > a_{\mathrm{IR}}$ in $d = 6$, or a unitary flow with $a_{\mathrm{UV}} \le a_{\mathrm{IR}}$.
Status in the literature
Unverified note
Proven for supersymmetric tensor-branch flows and, in 2023-2025, orbi-instanton Higgs-branch flows; open in general.