Hamilton's closure symplectic cone conjecture for minimal Kähler surfaces

Let $M$ underlie a minimal Kähler surface with $b^+>1$. Let $K$ be its canonical class, let $\mathcal P_M^K$ and $\mathcal P_M^{-K}$ be the cones of classes in $\mathcal P_M$ evaluating positively on $K$ and $-K$, respectively, and let $\overline{\mathcal C}_M$ be the closure of the symplectic cone in $H^2(M,\mathbb R)$.

Hamilton's closure symplectic cone conjecture.

$$\mathcal P_M^K\cup\mathcal P_M^{-K}\subset\overline{\mathcal C}_M.$$

This is a weaker version of the preceding symplectic cone conjecture: it asserts inclusion only in the closure of the symplectic cone. The source gives no resolution.