Consider a pure-birth process $X(t)$ with rates $\lambda_i$ that satisfies $\sum_{i=0}^\infty \frac{1}{\lambda_i} = \infty.$ By Reuter's criterion this is sufficient for $X(t)$ to be regular, ie $X(t) < \infty$ for all $t \ge 0$ holds a.s.
For $\lambda > 0$ let $\hat{X}(\lambda) := \int_0^\infty \lambda e^{-\lambda t} X(t) dt$ be the formal Laplace-Transform of $X(t)$.
Suppose there is a $\lambda^* > 0$ so that \begin{align} E\hat{X}(\lambda^*) &= 1 \\ E\int_0^\infty t e^{-\lambda^* t} X(t) dt & < \infty \end{align}
holds. Is the expected number of jumps $EX(a) \le 1$ for some $a$?