Existence of a weak Fano model for the tangent-bundle pair

Establish whether the pair consisting of the projectivized canonical extension and the projectivized tangent bundle of a ruled surface over an elliptic curve admits a weak Fano model obtained through a sequence of anti-flips, with the transformed projectivized tangent bundle a nef and big divisor.

Background

A proposed strategy for determining Steinness of the canonical extension is to construct a weak Fano model of the pair ((V^\omega),(T_M)). Such a model would involve anti-flips leading to a mildly singular variety on which the transformed projectivized tangent bundle is nef and big.

For the ruled surfaces considered in the paper, finite generation of the symmetric-tensor ring would be needed for this approach. The paper proves that the ring R(M,T_M) is not finitely generated and therefore explains why this strategy cannot be used in the present case; the existence of the weak Fano model itself is explicitly described as unclear.