Kollár's realization and exclusion conjectures for real components of uniruled varieties

Let $M$ be a closed orientable $3$-manifold. A Seifert fibered manifold is a manifold admitting a differentiable foliation by circles, and a lens space is a manifold diffeomorphic to a quotient of the $3$-sphere $S^3$ by the action of a cyclic group. Also consider the finite 0a priori0a list of exceptional manifolds arising in Kollár's classification and locally trivial torus bundles over $S^1$.

Kollár's conjectures.

1. If $M$ is an orientable Seifert fibered manifold, then there is a uniruled algebraic variety $X$ such that $M$ is diffeomorphic to a connected component of $X(\mathbb R)$.
2. If $M$ is a connected sum of lens spaces, then there is a uniruled algebraic variety $X$ such that $M$ is diffeomorphic to a connected component of $X(\mathbb R)$.
3. If $M$ belongs to the a priori given finite list of exceptional manifolds, or is a locally trivial torus bundle over $S^1$ that is not Seifert fibered, then $M$ is not diffeomorphic to a real component of a uniruled algebraic variety $X$.

These conjectures aim to complete Kollár's classification of orientable $3$-manifolds that can occur as connected components of real loci of uniruled algebraic varieties. The first two assertions predict realization, while the third predicts exclusion of the remaining exceptional cases and non-Seifert torus bundles; the supplied source gives no resolution status.