Convexity of the persistence exponent for fractionally integrated Lévy processes

For each $\beta\ge0$, let $\theta(\beta)$ denote the persistence exponent of a $\beta$-fractionally integrated Lévy process, when defined. The convexity conjecture. The function

$$\beta\mapsto\theta(\beta)$$

is convex decreasing.

The paper notes that existence is itself not established for every parameter range, and presents this shape property as a conjecture.