  A key fact about M.-P. Sch\"{u}tzenberger's (1972) promotion operator on
rectangular standard Young tableaux is that iterating promotion once per entry
recovers the original tableau. For tableaux with strictly increasing rows and
columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin
with applications to $K$-theoretic Schubert calculus. The author (2014) studied
a $K$-promotion operator $\mathcal{P}$ derived from this theory, but showed
that the key fact does not generally extend to $K$-promotion of such increasing
tableaux.
  Here we show that the key fact holds for labels on the boundary of the
rectangle. That is, for $T$ a rectanglar increasing tableau with entries
bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$,
where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last
row and column. Using this fact, we obtain a family of homomesy results on the
average value of certain statistics over $K$-promotion orbits, extending a
$2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary
rectangular shapes.
