Conjecture on semisimplicity criteria for Brauer algebras

Let $\kappa$ be a field, let $\hat{z}\in\kappa$, let $B_n(\hat{z})$ be an algebra over $\kappa$, and let $n\geq 2$. Let $\kappa\mathfrak{S}_n$ denote the group algebra of the symmetric group over $\kappa$, and let $p_i(z)$ be the polynomials defined by

$$p_1(z)=(z+2)(z-1),$$

with

$$p_i(z)=\begin{cases}(z+2i)(z-i)(z+i-2)p_{i-1}(z)&\text{if }i\text{ is odd},\\(z+2i)(z-i)p_{i-1}(z)&\text{if }i\text{ is even}. \end{cases}$$

Semisimplicity conjecture. (i) If $n=2k+1$, then $B_n(\hat{z})$ is semisimple only if $\kappa\mathfrak{S}_n$ is semisimple and $p_{2k-1}(\hat{z})\ne 0$. (ii) If $n=2k$, then $B_n(\hat{z})$ is semisimple only if $\kappa\mathfrak{S}_n$ is semisimple, $\hat{z}\ne 0$, and $p_{2k-2}(\hat{z})\ne 0$.

These are necessary conditions for semisimplicity of Brauer algebras, refining the preceding sufficient-condition discussion. The supplied text gives no resolution of this conjectural formulation.