Vanishing conjecture for coefficients in equivariant heat trace expansions

Let $\Gamma$ be a compact group acting isometrically and effectively on a compact, connected Riemannian manifold $M$. Let $a_{jk}$ denote the coefficients in the equivariant trace formula.

Equivariant heat coefficient vanishing conjecture. The coefficients satisfy

$$a_{jk}=0 \quad\text{for } k>0.$$

If $M$ is oriented and $\Gamma$ acts by orientation-preserving isometries, then

$$a_{j0}=0 \quad\text{for } j \text{ odd}.$$

The conjecture predicts that all logarithmic contributions to the equivariant heat trace vanish, and that the remaining coefficients have an additional parity vanishing under orientation-preserving symmetries. The paper notes that no examples with nonzero logarithmic terms are known, but does not establish the conjecture in general.