Connected sum of manifolds with spectral Ricci lower bounds

Let $n > 2$, γ > n-1/n-2, and λ ∈ R. We prove that if M and N are two smooth n-manifolds that admit a complete Riemannian metric satisfying \[ -γΔ + Ric > λ, \] then the connected sum $M # N$ also admits such a metric. The construction geometrically resembles a Gromov-Lawson tunnel; the range γ > n-1/n-2 is sharp for this to hold.