  Given a finite depth subfactor planar algebra $\mathcal{P}$ endowed with the
graded $*$-algebra structures $\{Gr_k^+ \mathcal{P}\}_{k\in\mathbb{N}}$ of
Guionnet, Jones, and Shlyakhtenko, there is a sequence of canonical traces
$Tr_{k,+}$ on $Gr_k^+\mathcal{P}$ induced by the Temperley-Lieb diagrams and a
sequence of trace-preserving embeddings into the bounded operators on a Hilbert
space. Via these embeddings the $*$-algebras $\{Gr_k^+\mathcal{P}\}_{k\in
\mathbb{N}}$ generate a tower of non-commutative probability spaces
$\{M_{k,+}\}_{k\in\mathbb{N}}$ whose inclusions recover $\mathcal{P}$ as its
standard invariant. We show that traces $Tr_{k,+}^{(v)}$ induced by certain
small perturbations of the Temperley-Lieb diagrams yield trace-preserving
embeddings of $Gr_k^+\mathcal{P}$ that generate the same tower
$\{M_{k,+}\}_{k\in\mathbb{N}}$.
