An $F$-compactum or a Fedorchuk compactum is a compact Hausdorff topological space that admits a decomposition into a special fully ordered inverse spectrum with fully closed neighboring projections. $F$-compacta of spectral height $3$ are exactly nonmetrizable compacta that admit a fully closed mapping onto a metric compactum with metrizable fibers. In this paper, it is proved that such a fully closed mapping for an $F$-compactum $X$ of spectral height $3$ is defined almost uniquely. Namely, nontrivial fibers of any two fully closed mapping of $X$ into metric compacts with metrizable inverse images of points coincide everywhere, with a possible exception of a countable family of elements. Examples of $F$-compacta of spectral height $3$ are, for example, Aleksandrov’s "two arrows" and the lexicographic square of the segment. It follows from the main result of this paper that almost all non-trivial layers of any admissible fully closed mapping are colons that are glued together under the standard projection of $D$ onto the segment. Similarly, almost all nontrivial fibers of any admissible fully closed mapping necessarily coincide with the "vertical segments" of the lexicographic square.