Does Ord essentially faint imply the eventual Singular Cardinal Hypothesis?

Ascertain whether the principle that the class of ordinals Ord is essentially faint implies the eventual Singular Cardinal Hypothesis; that is, whether for all sufficiently large singular cardinals κ, if 2^κ < κ^+, then κ^{cf(κ)} = κ^+ holds under the assumption that Ord is essentially faint.

Background

The paper shows that, over ZF together with the assumption that the Singular Cardinal Hypothesis (SCH) eventually holds, Ord essentially faint is equivalent to the existence of weak Löwenheim–Skolem–Tarski numbers for all abstract logics (Corollary SCHweakLST).

A full, unconditional equivalence would follow if Ord essentially faint entailed the eventual SCH. The authors explicitly note that it is not clear whether this fragment of SCH follows from Ord essentially faint, leaving this as an open question.