  Monotone inclusion problems occur in many areas of optimization and
variational analysis. Splitting methods, which utilize resolvents or proximal
mappings of the underlying operators, are often applied to solve these
problems. In 2022, Bredies, Chenchene, Lorenz, and Naldi introduced a new
elegant algorithmic framework that encompasses various well known algorithms
including Douglas-Rachford and Chambolle-Pock. They obtained powerful weak and
strong convergence results, where the latter type relies on additional strong
monotonicity assumptions. In this paper, we complement the analysis by Bredies
et al. by relating the projections of the fixed point sets of the underlying
operators that generate the (reduced and original) preconditioned proximal
point sequences. We also obtain strong convergence results in the case of
linear relations. Various examples are provided to illustrate the applicability
of our results.
