Connected sum of manifolds with spectral Ricci lower bounds

Let $n>2, \gamma>(n-1)/(n-2)$, and $\lambda$ a real number. Let $\mathrm{Ric}$ denote the lowest eigenvalue of the Ricci tensor. I will show that if $M$ and $N$ are two smooth n-dimensional manifolds that admit a complete Riemannian metric satisfying $-\gamma\Delta+\mathrm{Ric}>\lambda$, then their connected sum also admits a metric with the same property.
The construction is geometrically similar to a Gromov-Lawson tunnel, and the range $\gamma>(n-1)/(n-2)$ is sharp for this result to hold. Join work with K. Xu.