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We show that every finite GK-dimensional pre-Nichols algebra for braidings ofdiagonal type with connected diagram of modular, supermodular or unidentifiedtype is a quotient of the distinguished pre-Nichols algebra introduced by thefirst-named author, up to two exceptions. For both of these exceptional cases,we provide a pre-Nichols algebra that substitutes the distinguished one in thesense that it projects onto all finite GK-dimensional pre-Nichols algebras. Webuild these two substitutes as non-trivial central extensions with finiteGK-dimension of the corresponding distinguished pre-Nichols algebra. Wedescribe these algebras by generators and relations, and provide a basis.  This work essentially completes the study of eminent pre-Nichols algebras ofdiagonal type with connected diagram and finite-dimensional Nichols algebra.

Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type