Sharpness conjecture for the lower semigroup exponent of complex elliptic operators

Let $n\ge 3$, let $\mathcal{E}$ be the class of all complex elliptic operators under study, and for $L\in\mathcal{E}$ let $p_-(L)$ be the lower endpoint of the maximal interval of exponents $p$ for which the semigroup $(e^{-tL})_{t>0}$ is $L^p$ bounded. Sharpness conjecture. The inequality

$$p_-(L)<\frac{2n}{n+2}$$

is sharp for the class $\mathcal{E}$; equivalently, for every $p<\frac{2n}{n+2}$ there exists $L\in\mathcal{E}$ such that $p_-(L)>p$. The known upper bound is therefore conjectured to be optimal, but the existence of such operators for every $p<\frac{2n}{n+2}$ remains open.