Proposition: A unit gap above a degenerate ground space permits volume-law ground vectors

There are qubit Hamiltonians on even open $L\times L$ squares with $R=0$, $J=1$, and gap one above the ground space, containing ground vectors with

Thus the extension to every vector of an unrestricted degenerate ground space is false.

Proof. Let $A$ be the left half of the square. Pin one site $x_0\in A^c$ with

The spectrum is $\{0,1\}$, with a highly degenerate ground space.

Choose $m=L^2/2-1$ disjoint pairs joining distinct sites of $A$ to distinct sites of $A^c\setminus \{x_0\}$. Put a Bell pair on each pair, and put $|0⟩$ on the two unpaired sites, including $x_0$. This is a ground vector. Each Bell pair contributes $\log 2$ across the vertical cut.