Is self-correction impossible for every two-dimensional local Hamiltonian
In plain words
Simple two-dimensional quantum codes are proven unable to protect a qubit passively against heat. Whether every two-dimensional system, including more complicated ones, must also fail is not proven.
Precise statement
For any 2D local Hamiltonian on finite-dimensional spins (commuting or not, stabilizer or not) coupled weakly to a thermal bath at $T > 0$, prove that the memory time of every encoded qubit is bounded by a function of $T$ alone, independent of system size, or give a counterexample. No-go theorems cover 2D commuting Pauli stabilizer codes (Bravyi and Terhal 2009) and commuting-projector models with local topological order (Landon-Cardinal and Poulin 2013).
What would settle it
A general no-go proof for $2\mathrm{D}$ local Hamiltonians, or an explicit $2\mathrm{D}$ model with memory time growing with size at fixed $T > 0$.