Regularity of the growth of Dirichlet series with respect to a strongly incomplete exponential system

The article deals with the behavior of the sum of the Dirichlet series $F(s)=\sum\nolimits_{n} a_ne^{\lambda_ns}$, with $0<\lambda_{n}\uparrow\infty$, converging absolutely in the left half-plane $\Pi_0=\{ s=\sigma+it: \sigma<0\}$ along a curve arbitrarily approaching the imaginary axis, the boundary of this half-plane. We assume that the maximal term of the series satisfies some lower estimate on some sequence of points $ \sigma_n \uparrow 0-$. The essence of the questions we consider is as follows: Given a curve $\gamma$ starting from the half-plane $\Pi_0$ and ending asymptotically approaching on the boundary of $\Pi_0$, what are the conditions for the existence of a sequence $ \{ \xi_n\} \subset\gamma$, with $\operatorname{Re} \xi_n \to 0-$, such that $ \ln M_F(\operatorname{Re} \xi_n) \sim \ln\vert F(\xi_n)\vert $, where $M_F(\sigma)=\sup\nolimits_{\vert t\vert <\infty}\vert F(\sigma+it) \vert $? A.M. Gaisin obtained the answer to this question in 2003. In the present article, we solve the following problem: Under what additional conditions on $\gamma$ is the finer asymptotic relation valid in the case that the argument $s$ tends to the imaginary axis along $\gamma$ over a sufficiently massive set?