  Since the discovery of the famous Ryu-Takayanagi formula, the AdS/CFT
correspondence has seen a surge in interest from the quantum-information
community. Tensor networks have emerged in parallel as a useful tool for
simulating certain aspects of this correspondence. Two of these ans\"atze, the
so-called "perfect tensor networks" (or HaPPY) and the "multiscale entanglement
renormalization ansatz" (or MERA), have become paradigms of a field, realizing
distinct but related aspects of AdS/CFT. Recently, a tensor-network ansatz was
introduced which combines the advantages of both of these ans\"atze. These
networks, known as "hyperinvariant tensor networks", have not been demonstrated
as a viable tensor-network class due to several practical considerations, most
of which are related to issues of boundary quasiperiodicity which arise in
tensor networks based on regular tessellations of hyperbolic manifolds. In this
paper, we discuss these challenges with the aim of explaining current
difficulties on more technical grounds; in the process, we propose two new
tensor decompositions for hMERA that include antisymmetric tensors,
demonstrating a potential flexibility for describing physical systems in the
network. Additionally, we provide numerical evidence which suggests that the
Ising CFT could be reliably captured, provided that modifications are made to
the current tensor decompositions in both the {7,3}- and
{5,4}-parameterizations. We conclude by analyzing the current tensor
decompositions in hyperinvariant tensor networks. This analysis is anticipated
to aid in the simulation of conformal field theories in tensor networks which
are based on regular tessellations of negative-curvature hyperbolic manifolds.
