The claim is not that Articulatory Phonology "supports" Autosegmental Phonology, rather, it is a factual recognition that ArtPho has a certain resemblance to AutoPho. The graphic representation is introduced in Goldsmith's dissertation in (2) of ch. 1, where he gives the "score for the orchestration of 'pin'", contrasting the standard segmental view with a physical "score". Chapter 4 then talks about "laryngeal gestures". B&G pick up from that perspective, and try to say in detail how this kind of graph can be related to actual physical measurements.
As Goldsmith points out (ch. 4), without autosegmental phonology, the linguistic existence of segments is axiomatic, but with autosegmental phonology, it becomes an empirical hypothesis. The reason is that Goldsmith's gestural score is also a conceivable physical output, so the obvious question to ask is whether something else (other than sheer logical necessity, the need to have a way to talk about what is produced) motivates having segments. Phonologists generally hold that there is something – grammatical computations – that do motivate segments. The ArtPhon program has been to subsume as much of phonology as possible under the kind of "gesture-sliding" analysis that they give in that and other papers. (For example reanalyzing apparent category-changing assimilations of /t/ to [p] before [p] not as change of thing, but an earlier initiation of labial closure which obscures the lingual gesture). There is, not entirely incidentally, a de-linguistification, of phonology in ArtPho, where the goal is to reduce at least phonetic implementation to a general motor control system, similar to walking or chewing.