Classical efficiency of 2D gapped ground-state energy estimation

Is there a classical algorithm that, for every $2\mathrm{D}$ nearest-neighbour Hamiltonian on $n$ qudits with spectral gap $\Delta = \Omega(1)$, outputs the ground energy to additive error $1/\operatorname{poly}(n)$ in time $\operatorname{poly}(n)$? The $1\mathrm{D}$ case is in $\mathrm{P}$ (Landau, Vazirani and Vidick 2015); without the gap promise the $2\mathrm{D}$ problem is QMA-complete.

A polynomial-time algorithm with proof, or a hardness result for the gapped 2D promise problem.