Strict cone conjecture for the $L^p$-spectrum of the Laplacian on forms

Let $M$ be a complete noncompact Riemannian manifold with finite exponential rate of volume growth $\gamma<\infty$. Let $\Delta$ denote the Laplacian on $k$-forms, and assume that the Weitzenböck curvature on $k$-forms has a lower bound for some $0\leq k\leq n$. For a fixed $p$, write $L^p(\Lambda^k(M))$ for the space of $p$-integrable differential $k$-forms on $M$. Strict cone conjecture. There exist real numbers $c\leq 0$ and $a>0$ such that, for every complex number $w$ satisfying

$$\operatorname{Re}(w)\leq a|\operatorname{Im}(w)|-c,$$

the resolvent $(\Delta-w)^{-1}$ is bounded on $L^p(\Lambda^k(M))$. The paper proves this spectral inclusion under the stated curvature assumptions in the special case discussed, while the general conjectural strict-cone phenomenon is motivated by earlier results on conical enclosures of the $L^p$-spectrum.