Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.

Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:

If $U(a,b)$ denotes the number of up-crossings of $X$ through $(a,b)$ up to time $1$, then

$$\mathbb E \big[U(a,b)\big] \leq \frac{1}{b-a}\mathbb E\big[(a-X_1)^+\big].$$

This inequality should essentially never be sharp for a martingale with continuous sample paths, except for those almost surely constant in time, or those that never hit $a$ to begin with.

Suppose now $X$ is a Brownian motion. How much can we sharpen this bound?