Tightness of extremal points of discrete bridge arches to the Brownian limit

Let $(S_k)_{0\leq k\leq n}$ be a discrete bridge formed from centered, variance-one i.i.d. increments with an exponential moment, and consider sufficiently long positive or negative arches of the bridge. The corresponding Brownian limit is the standard Brownian motion on the unit interval, with heights normalized by the usual factor of the bridge length.

Tightness conjecture. The highest points of sufficiently long positive arches, and the lowest points of sufficiently long negative arches, are tight to the Brownian limit.

The preceding discussion compares the discrete bridge with Brownian motion through a strong approximation and identifies the Brownian extremal-height distribution. The conjecture asserts that the extremal points of long arches inherit this limiting tightness, beyond the pointwise approximation currently established.