There are no official opinion polls that address the question, but based on my knowledge of contemporary phonology and phonetics, and phonologists and phoneticians, there is (now) a higher percentage of phonologists who believe in syllables than phoneticians. The premise of there being a "phonetic syllable" is in serious doubt. There are opposite perspectives on the topic within the realm of phoneticians, so John Ohala is a non-supporter of the syllable and Osamu Fujimura is a strong supporter. In addition, there is the well-known huge variation in what constitutes "phonetic" versus "phonological" analysis, with the poles being roughly "the tail end of the phonology" versus "articulatory movements and waveforms". So without staking out some more specific territory, there is no way to decide on an answer.
The standard methodological approach to making scientific decisions favors the conclusion that syllables are not necessary to explain the facts, but if they are, they are phonological objects and not phonetic objects. There is no such thing as a phonetic computation of syllable parsing, in the way that one can compute formants, spectral tilt, amplitude and F0. As is the case in phonology, claiming that a certain string of segments is parsed into syllables as [abc.def] as opposed to [ab.cdef] or [abcd.ef] is based on the simplicitity and utility of performing that parse. There is a certain amount of evidence for the syllable based on phonological considerations such as rules which apply to segment sequences within the same syllable -- for example, velar palatalization in Turkish has been claimed (Clements & Sezer) to be syllable conditioned. English consonant allophony has been analyzed as being conditioned by syllable position, e.g. in Kahn's dissertation, but it has also been analyzed as being sensitive to foot position. Theoretically, the case for syllables at the phonetic level could be made in a similar fashion, but the difficulty lies in a chicken / egg question whether syllabification contrasts are primary and observed segmental differences derive from phonetic rules referring to the syllable, or are the segmental differences already there in the output of the phonology, in which case the phonetics has no need to refer to syllables. That is, parsing segments into syllables has no utility in phonetics (or, that would be the claim).
As a rule of thumb, if the facts that you're trying to describe require you to refer to a continuum of values, then you're dealing with a phonetic fact. If you have such evidence that appears to require reference to the syllable, then a case can be made that this is a syllable-sensitive phonetic process, since (as a continuous as opposed to discrete process) the facts can't be disposed of by relegating the process to phonology, where the case for the syllable is stronger.