Gnomad:
I was told long ago that the longest English consonant string in an actual word (though admittedly one originally made by joining two together) is Knightsbridge.

That's a string of 6 consonants consecutively, in a word you admit is a construct.
rhythms has 7 and is a word in general use. Admittedly the 'y' seems to be being used as a vowel, but by most definitions it beats Knightsbridge.
Knightsbridge can also be equalled by "catchphrase"

Cheers
Nigel