In Del Pinal, Bassi, Sauerland (2024), we present a novel Grammatical theory of FC based on a presuppositional exhaustification operation, called 'pex', and show that it resolves various puzzles concerning the way in which FC and related sentences project from various embedded environments. In that paper, we derive basic FC effects by assuming that 'pex' is sensitive to both innocently excludable (IE) and includable (II) alternatives (call this 'pex^IE+II'). This note is intended as an addendum to our main paper: we show that we can derive the same basic results using a purely IE-based presupositional exhaustification operator (call this 'pex^IE'). Relative to the target readings, 'pex^IE+II' and 'pex}^IE' are descriptively on equal footing. Yet there are non-trivial theoretical trade-offs: while the derivation of basic FC effects is simpler with 'pex^IE+II'---in that it doesn't require recursive exhaustification---'pex^IE' uses simpler operations which are sensitive only to IE alternatives. In this note, we present two models to derive FC effects with 'pex^IE': one appeals to a local accommodation operator, and the other on a deeper exploitation of Strong Kleene semantics.