here numberings are as in the update version.
a precise link to a section of [54] with a calculation indicative of the
`routine calculation' omitted in the proof of 4.26 is here:
lecture notes Ch.15
(This is a chapter on the ordinary partition algebra. But the argument
for the Kadar-Yu algebras is directly analogous.
Section numbers in the lecture notes are not super-stable, since the
notes vary from year to year, but currently the Lemma number is around 15.4.10.)