  We compute connection probabilities for reduced $3$-webs in the triple-dimer
model on circular planar graphs using the boundary measurement matrix (reduced
Kasteleyn matrix). As one application we compute several "$\text{SL}_3$
generalizations'' of the Lindstr{\o}m-Gessel-Viennot theorem, for "parallel"
webs and for honeycomb webs. We also apply our results to the scaling limit of
the dimer model in a planar domain, giving conformally invariant expressions
for reduced web probabilities.
