Failure of semipositive metric extension in a smooth projective family

We construct a smooth projective family of rational surfaces, an effective line bundle on its total space, and a prescribed semipositively curved singular Hermitian metric on the central fibre that has no semipositive extension to any neighbourhood of that fibre. It has analytic singularities and is of minimal singularity type. The failure persists even when the restriction is only required to have the same singularity type as the prescribed metric. The family is obtained by blowing up four disjoint sections of a product, with three points becoming collinear on the central fibre. An integrable adjoint section on that fibre cannot extend because the corresponding adjoint systems vanish on every nearby fibre. This gives a negative answer to Păun’s metric-extension question, listed as Question 38 by Dinew, Guedj, and Zeriahi.