How many readings does (1) have?

(1) John washed his car and Bill waxed it.

Gerhard Jäger says it has two:

(2) John washed John’s car and Bill waxed John’s car.
(3) John washed John’s car and Bill waxed Bill’s car.

Indeed, he uses this as a paradigm case of a ‘sloppy identity’ reading for anaphoric pronouns. But I find it very hard to hear (1) as meaning (3). Am I just missing something here?

I do think there are cases where we get these ‘sloppy’ anaphora, as in classic examples like

(4) The man who gave his paycheck to his mistress was smarter than the man who gave it to his bookie.

But it’s worth noting that the antecedent (is that the right word?) of the anaphora is bound by a quantifier here, not a name. I can’t off the top of my head think of a case like (1) where the initial phrase ‘his F’ is bound to a name, and the ‘it’ picks up the sense (as it were) of that phrase not its reference.