[M94]
In Sec.6.2.2, N.B., the equivalence is not a congruence,
so $P_n^{\sim}(Q) = KS_n^{\sim}$ (writing $S_n^{\sim}$ for the
set of classes of $S_n$ under $\sim$) is not a quotient algebra,
and the construction should be understood combinatorially.
(With the benefit of hindsight) A good way to notate this
is in the categorical terms introduced in Sec.7.
In consequence Prop.12 concerns representations of $P_n(Q)$.
The Outline proof of Prop.12 is somewhat terse. Since not every
row of the gram matrix is diagonal dominated one needs to note
the partial order on basis elements by the number of isolated
parts. Each row is diagonal dominated if one ignores columns
labelled by elements higher in this partial order. Now consider
the Laplace expansion.
[MS93] contains a version of the same construct $\sim$ as in [M94,Sec.6.2.2]
above , which can be understood in the same way.