The phonetic transcription for desirous (in its BrE pronunciation) can be written as [dɪˈzʌɪərəs]. I wonder which diphthong—ʌɪ or ɪə—takes place in this word? Maybe here we can see a kind of haplology?
/ʌɪə/ and /ʌuə/ are triphthongs. They are sometimes phonetically realized as diphthongs, but if all three vowel sounds are pronounced, they are triphthongs.
In my pretty standard version of BrE it is not a tripthong and I've never heard it that way.
[dɪˈzʌɪə] - tripthong, emphasis on ʌɪ, silent r
[dɪˈzʌɪrəs] - dipthong