Optimal Tick Improvement for a Market Maker

A market maker can improve their quote by $k \in \{0, 1, 2, \ldots\}$ ticks (each tick has size $\tau$). Improving by $k$ ticks multiplies the conditional-on-queue fill intensity by $e^{\lambda k}$ for some $\lambda > 0$, but reduces the per-fill spread capture from $s$ to $s - k\tau$. Ignoring adverse selection differences across $k$, choose the optimal tick improvement $k^{*}$ that maximizes expected profit per unit time. Under what condition on $s$, $\lambda$, and $\tau$ is it optimal to improve by at least one tick?