  Motivated by the problem of finding constant scalar curvature K\"ahler
metrics, we investigate a Ricci iteration sequence of Rubinstein that
discretizes the pseudo-Calabi flow. While the long time existence of the flow
is still an open question, we show that the iteration sequence does exist for
all steps, along which the K-energy decreases. We further show that the
iteration sequence, modulo automorphisms, converges smoothly to a constant
scalar curvature K\"ahler metric if there is one, thus confirming a conjecture
of Rubinstein from 2007 and extending results of Darvas--Rubinstein to
arbitrary K\"ahler classes.
