The absence of CLP realizations for chiral topological phases
A chiral topological phase of matter (TPM) is a TPM whose topological order is a chiral unitary modular category. A CLP Hamiltonian is a commuting local-projector Hamiltonian. Chiral CLP non-realization conjecture. There are no CLP Hamiltonian realizations of chiral TPMs. The source says that numerical simulations suggest examples of chiral TPMs, but rigorous proofs for the relevant non-CLP Hamiltonians are out of reach.
References
Primary source
Eric C. Rowell and Zhenghan Wang, “Mathematics of Topological Quantum Computing”, arXiv:1705.06206 (2017).
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