complete intersection on $\mathbb{P}^1\times\mathbb{P}^1$?

Let $S=k[x_0,x_1;y_0,y_1]$ be the bihomogeneous coordinate ring of $\mathbb{P}^1\times\mathbb{P}^1$. Suppose that $F\in S_{(d_1,d_2)}$ is a generic bihomogeneous form of bidegree $(d_1,d_2)$, and let $F_1\in S_{(a_1,b_1)}$, $F_2\in S_{(a_2,b_2)}$ be generic forms of lower bidegrees.

The question is when $F\in (F_1,F_2)$.

Geometrically, this asks when a generic divisor of type $(d_1,d_2)$ on $\mathbb{P}^1\times\mathbb{P}^1$ contains the complete intersection $V(F_1,F_2)$.

In Complete Intersections on General Hypersurfaces, the authors study the analogous question in projective space. Is there an analogous result for $\mathbb{P}^1\times\mathbb{P}^1$?