The strict upper bound for the distinguished ECH spectrum

Let $(Y,\lambda)$ be a closed contact $3$-manifold satisfying the assumptions of the distinguished ECH volume conjecture, namely that it has nonvanishing ECH contact invariant and $c_k(Y,\lambda)<\infty$ for all $k$.

Strict ECH volume bound.

$$c_k(Y,\lambda)<\sqrt{2k\operatorname{vol}(Y,\lambda)}\qquad\text{for all }k>0.$$

This conjecture is a proposed refinement of the asymptotic volume formula: it gives a strict upper bound at every finite index, whereas the volume conjecture concerns the limiting ratio. The source reports only limited experimental support, and no general proof is given.