Does the a-anomaly decrease along all six-dimensional RG flows?

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.

A general argument (dilaton, entanglement or other) establishing $a_{\mathrm{UV}} > a_{\mathrm{IR}}$ in $d = 6$, or a unitary flow with $a_{\mathrm{UV}} \le a_{\mathrm{IR}}$.