\$2\$-ray game in dimension two

Let $$S$$ be a projective surface of Picard rank two over a field $$K$$. Assume that the cones of nef divisors and effective divisors on $$S$$ coincide, and that the effective cone is closed, that is all pseudo-effective divisors are $$mathbb{Q}$$-effective.

Denote by $$D_1,D_2$$ the divisors on $$S$$ generating the rays of $$text{Nef}(S) = text{Eff}(S)$$. Assume that $$D_1$$ yields a fibration $$Srightarrow mathbb{P}^1$$ whose general fiber is a rational curve. Could the map induced by $$mD_2$$ be constant (meaning that it contracts $$S$$ to a point) for all $$mgeq 1$$ or is such a contraction forced to have relative Picard rank one?