Finite transit-time conjecture for mean-median orbits

Let an initial set be a finite multiset of real numbers, and iterate the mean-median map, which adjoins at each step a real number so that the arithmetic mean of the enlarged set equals the median of the preceding set. The resulting sequence of newly generated numbers is the orbit's sequence of iterates.

Finite transit-time conjecture. For every initial set, the sequence of new numbers generated by iterating the mean-median map is eventually constant.

This conjecture asserts that every mean-median orbit has finite transit time, meaning that it eventually becomes stationary. It has been formulated over the rationals and remains open, although the paper establishes families of initial sets whose transit times grow linearly and quadratically with the size of the set.