The refined Sharp Kahane conjecture for ordinary Dirichlet series

Let $L(s)=\sum a_n n^{-s}$ be an ordinary Dirichlet series with entire continuation of order at most $1$, and let $\mu(\sigma)$ denote its order function on vertical lines. Refined Sharp Kahane conjecture. For every $\sigma$ satisfying $\mu(\sigma)\ge 1$,

$$\mu\bigl(\sigma+\mu(\sigma)\bigr)=0.$$

Equivalently, on any interval where $\mu\ge 1$, the slope of $\mu$ is at most $-1$. The unrestricted relation $\mu(\sigma+\mu(\sigma))=0$ would imply the Lindelöf hypothesis for the Riemann zeta function, so the paper isolates this restricted form as a potentially accessible statement; its status is presented as open.