Recursive construction of central idempotents in partition algebras

Let $D_r$ and $D'_r$ be the partition-algebra objects containing the rank-one central idempotents $E(r)$ and $E'(r)$, respectively. Let $h_i$, $s_i$, and $p_i$ denote the generators defined by the corresponding partition diagrams, and let $\rho$ be the one-dimensional representation used to define these idempotents. Recursive idempotent construction. The idempotents $E(r)\in D_r$ and $E'(r)\in D'_r$ are constructed recursively by $E'(1)=1$ and

$$E(r) = \frac1r E'(r)\left[ 1+(r-1)s_{r-1} -\frac1{\delta-2r+2}p_r \right] E'(r)$$ $$E'(r+1) = E(r)\left[ 1 - r\left(\frac{\delta-2r+2}{\delta-2r+1}\right)h_r \right] E(r).$$

This gives an explicit recursive description of the central idempotents associated with the one-dimensional representations of the partition algebras. The source presents the assertion as a conjecture, and no resolution is supplied here.