$2$-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?