Conjecture on non-degeneracy of bilinear forms for Brauer algebra modules

Let $\kappa$ be a field, let $\hat{z}\in\kappa$, and let $B_n(\hat{z})$ be an algebra over $\kappa$, with $\geq 2$. Define polynomials $p_i(z)$ by $p_1(z)=(z+2)(z-1)$ and

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

Non-degeneracy conjecture. (i) If $n=2k+1$, then the bilinear form on the $B_n(\hat{z})$-module $S^{(1)}$ determined by the referenced construction is non-degenerate if and only if $p_k(\hat{z})\ne 0$. (ii) If $n=2k$, then the bilinear form on the $B_n(\hat{z})$-module $S^{\varnothing}$ determined by the referenced construction is non-degenerate if and only if $\hat{z}\ne 0$ and $p_k(\hat{z})\ne 0$.

These criteria give explicit parameter conditions for non-degeneracy of bilinear forms on the indicated cell modules of Brauer algebras. The surrounding discussion contrasts them with semisimplicity criteria, but the supplied text does not establish their resolution.