Do triphthongs exist?

Here I add another entry from John Wells's blog.
A very interesting issue.
And, as I always tell you, there is never A answer. Different very much respected phoneticians have their own view.

triphthongs, anyone?
In the comments on last Friday’s blog we touched on the issue of words like fire and power: are these monosyllables or disyllables? I mentioned that in LPD I treat them as varisyllabic.Let’s look at this more closely. Some authors describe the English vowel system as including not only diphthongs but also triphthongs. Peter Roach (English Phonetics and Phonology, 4th ed., CUP 2009, p. 18-19) puts it like this:
The most complex English sounds of the vowel type are the triphthongs. They can be rather difficult to pronounce, and very difficult to recognise. A triphthong is a glide from one vowel to another and then to a third, all produced rapidly and without interruption.
He lists the triphthongs eɪə, aɪə, ɔɪə, əʊə, aʊə (later giving the example words layer, player; liar, fire; loyal, royal; lower, mower; power, hour) and continues
The principal cause of difficulty for the foreign learner is that in present-day English the extent of the vowel movement is very small, except in very careful pronunciation. Because of this, the middle of the three vowel qualities of the triphthong (i.e. the ɪ or ʊ part) can hardly be heard and the resulting sound is difficult to distinguish from some of the diphthongs and long vowels. To add to the difficulty, there is also the problem of whether a triphthong is felt to contain one or two syllables. Words such as ‘fire’ or ‘hour’ are probably felt by most English speakers (with BBC pronunciation) to consist of only one syllable, whereas ‘player’ pleɪə or ‘slower’ sləʊə are more likely to be heard as two syllables.
I find this account unsatisfactory. If the əʊə of slower is a “triphthong”, it is difficult to see any reason why the əʊɪ of going is not one too. If liar has a triphthong, surely trying must have one.More to the point, a diphthong is not any old “movement or glide from one vowel to another” (Roach’s definition, p. 17). A word like neon niːɒn certainly has a movement or glide from one vowel to another, but does not contain a diphthong, because it has two syllables. A diphthong is a vowel glide within a single syllable. Ashby and Maidment (Introducing Phonetic Science, CUP 2005, p. 8) put it like this:
Since voice and house are one-syllable words, English diphthongs must count as one V element rather than two. Both voice and house have the structure CVC, rather than CVVC.
Similarly, I would argue that part of the definition of a true triphthong must be that it constitutes a single V unit, making with any associated consonants just a single syllable.Given that, do we have triphthongs in English? I claim that generally, at the phonetic level, we don’t. I treat the items we are discussing as basically sequences of a strong vowel plus a weak vowel. (By ‘strong vowel’ I mean one that is stressable and the potential input to a weakening rule; by ‘weak vowel’ I mean one that is the potential output of a weakening rule. Diphthongs such as aɪ are included under the heading ‘strong vowel’.)These sequences are subject to two possible processes: smoothing and compression. Smoothing means the loss of the second part of the strong vowel (diphthong). Compression means the squashing of the two syllables into one syllable. Both of these processes are optional (or stylistically determined).Hence given the disyllabic starting point paʊ.ə power, we can smooth it to disyllabic pa.ə. We can then compress the result to give monosyllabic paə. (This may be subject to the further process of Monophthonging, giving paː.) Similarly, ɡəʊ.ɪŋ going can be smoothed to ɡə.ɪŋ and then compressed to ɡəɪŋ.If my definition of triphthong holds, then a triphthong would be generated only if we apply Compression without first applying Smoothing. And my claim is that we do not commonly do that.