Separately omitting the Euler-characteristic hypotheses

Determine whether the Marian–Oprea strange duality isomorphism theorem remains valid when exactly one of the two sign conditions \(\chi(v)\le 0\) or \(\chi(w)\le 0\) is omitted while all other hypotheses of the theorem are retained.

Background

The paper’s Picard-rank-one counterexamples violate both the condition $\chi(w)\le0$ and the lower bound on $w^2$ simultaneously. Consequently, those examples do not determine whether either sign condition can be removed on its own.

The unresolved problem is to isolate the effect of relaxing only one Euler-characteristic sign condition, while preserving the theorem’s remaining numerical and geometric assumptions.