  We further exploit the relation between tropical Grassmannians and
$\operatorname{Gr}(4,n)$ cluster algebras in order to make and refine
predictions for the singularities of scattering amplitudes in $\mathcal{N}=4$
planar super Yang-Mills theory at higher multiplicity $n\ge 8$. As a
mathematical foundation that provides access to square-root symbol letters in
principle for any $n$, we analyse infinite mutation sequences in cluster
algebras with general coefficients. First specialising our analysis to the
eight-particle amplitude, and comparing it with a recent, closely related
approach based on scattering diagrams, we find that the only additional letters
the latter provides are the two square roots associated to the four-mass box.
In combination with a tropical rule for selecting a finite subset of variables
of the infinite $\operatorname{Gr}(4,9)$ cluster algebra, we then apply our
results to obtain a collection of $3,078$ rational and $2,349$ square-root
letters expected to appear in the nine-particle amplitude. In particular these
contain the alphabet found in an explicit 2-loop NMHV symbol calculation at
this multiplicity.
