Directed landscape convergence for the half-space log-gamma polymer $N^{2\mathrm{/}3+\delta }$ away from the boundary

We prove that the free energy of the half-space log-gamma polymer $N^{2\mathrm{/}3+\delta }$ away from the boundary in the non-attractive regime converges to the directed landscape. Based on the convergence of the full-space log-gamma free energy to the directed landscape, we couple the full-space and the half-space model and prove that the dominant contributions to free energy in both cases come from paths that remain confined to a transversal window of order $N^{2\mathrm{/}3}$. The result follows from three main inputs: a deterministic leading-order gap between paths that deviate from the transversal window on the $N^{2\mathrm{/}3+\delta }$ scale and those within the typical $N^{2\mathrm{/}3}$ scale; uniform exponential upper-tail bounds for half-space free energies with general slope; and existing full-space estimates on constrained and exiting free energies.