Classification in the Z''13 family

Classify the integer quadruples $(a,b,c,d)$ for which the canonical bundle of the smooth proper toric threefold $Z''_{13}(a,b,c,d)$ is tensor-generating, and determine whether every obstruction throughout this family is an invariant $(-1,-1)$-curve.

Background

The paper constructs two infinite subfamilies of the smooth non-projective toric threefolds $Z''_{13}(a,b,c,d)$ whose canonical bundles are tensor-generating for specified parameter values. In one subfamily, the exceptional parameter values are explained by invariant obstructing curves of type $(-1,-1)$. The authors explicitly leave open the classification for all parameters and whether the same geometric form of obstruction persists throughout the entire family.