Tian–Jun Li's symplectic cone conjecture for minimal Kähler surfaces

Let $M$ underlie a minimal Kähler surface with $b^+>1$. Write $K$ for the canonical class, and let $\mathcal C_M$ be the symplectic cone. For a cohomology class $K$, let $\mathcal P_M^K$ denote the cone of classes in $\mathcal P_M$ that evaluate positively on $K$.

Tian–Jun Li's symplectic cone conjecture.

$$\mathcal P_M^K\cup\mathcal P_M^{-K}\subset\mathcal C_M.$$

Together with the known inclusion $\mathcal C_M\subset\mathcal P_M^K\cup\mathcal P_M^{-K}$, this would determine the symplectic cone for these surfaces. The source presents this as Question 4.9 and gives no resolution.