Equivalence between Ord essentially faint and weak LST numbers for all abstract logics

Determine whether the scheme asserting that every abstract logic has a weak Löwenheim–Skolem–Tarski number is equivalent to the principle that the class of ordinals Ord is essentially faint (equivalently, to the scheme asserting that every abstract logic has a strict Löwenheim–Skolem–Tarski number).

Background

The paper proves that Ord is essentially faint is equivalent to the existence of strict Löwenheim–Skolem–Tarski numbers for all abstract logics (Theorem MAIN:StrictLST).

It also gives a characterization of when every abstract logic has a weak Löwenheim–Skolem–Tarski number in terms of the existence, for each natural number n and cardinal η, of a C^n-weakly shrewd cardinal κ with strong closure properties under exponentiation (Theorem CharWeakLST).

Corollary SCHweakLST shows that, assuming the eventual Singular Cardinal Hypothesis (SCH), Ord essentially faint is equivalent to the existence of weak Löwenheim–Skolem–Tarski numbers for all abstract logics. The unconditional equivalence remains unsettled and is explicitly highlighted as unclear.