Convexity conjecture for the positive total Hofer norm

Let $(M,\omega)$ be the symplectic manifold under consideration and let $f\in \operatorname{Ham}(M,\omega)$. Define the positive total Hofer norm as the function assigning to each $n\in\mathbb{N}$ the coefficient $\|f\|_n^+$ of the polynomial $\|f\|_T^+=\sum_{k\geq 0}\|f\|_k^+t^k$. Convexity conjecture. For all $g>0$,

$$\|f\|_{g+1}^+-\|f\|_g^+\leq \|f\|_g^+-\|f\|_{g-1}^+.$$

The conjecture asserts a regularity property for the sequence of positive $g$-areas and is presented as the principal conjecture concerning their behaviour. Its status is not determined by the supplied text.