Triphthongs

Students asked whether they should shorten diphthongs followed by the schwa. They were not aware of the description of triphthongs.
Here I include Peter Roach's opinion with respect to this topic:
"A triphthong is a glide from one vowel to another and then to a third, all produced rapidly and without interruption. E.g. a careful pronunciation of the word 'hour' begins with a vowel quality similar to /a:/(long a), goes on to a glide towards the back close rounded area (short u), then ends with a mid-central vowel (schwa).
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 triphthng (that is the i or u 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 sucha as 'fire' or 'hour' are probably felt by most English speakers (with BBC pronunciation) to consist of only one syllable, whereas 'player' or 'slower' are more likely to be heard as two syllables."
(ROACH, P.(2000) English Phonetics and Phonology. A practical course. C.U.P. Cambridge)

In the Cambridge Pronouncing Dictionary the three symbols appear in the same syllable. In the Longman Pronunciation Dictionary, the diphthongs appear with a small arc linking them to the schwa, which means that they are pronounced together.

I could definitely assert that the schwa is not part of a separate syllable.

Hope it helps!