The sharp asymptotic bounds conjecture for the density of sections on Kähler manifolds

Let ${\mathcal K}(n,c,V)$ be the class of Kähler manifolds under consideration, let $D=D(c)$ be the integer defined above, and let $\rho_{kD,X}$ denote the corresponding density invariant for $X\in{\mathcal K}(n,c,V)$. For any $n,c,V$ and $\eta<1$, there is a number $k_{0}(n,c,V,\eta)$ such that, whenever $k\geq k_{0}$ and $X\in{\mathcal K}(n,c,V)$,

$$\eta (2\pi)^{-n} (kD)^{n}\leq \rho_{kD,X}\leq \eta^{-1} c^{-1}(2\pi)^{-n} (kD)^{n}.$$

Sharp asymptotic bounds conjecture. For any $n,c,V$ and $\eta<1$, there is a number $k_{0}(n,c,V,\eta)$ such that if $k\geq k_{0}$ then for any $X$ in ${\mathcal K}(n,c,V)$ the displayed two-sided bounds hold. This would substantially sharpen Theorem 1.1 by giving matching-order bounds for the density invariant uniformly over the class.