Connected sum of manifolds with spectral Ricci lower bounds

Let $n>2$, $\gamma >\frac{n-1}{n-2}$, and $\lambda \in \mathbb{R}$. We prove that if $M$ and $N$ are two smooth $n$-manifolds that admit a complete Riemannian metric satisfying $$-\gamma \mathrm{\Delta }+\mathrm{R}\mathrm{i}\mathrm{c}>\lambda ,$$ then the connected sum $M\mathrm{\#}N$ also admits such a metric. The construction geometrically resembles a Gromov-Lawson tunnel; the range $\gamma >\frac{n-1}{n-2}$ is sharp for this to hold.