Even-or-odd invariant forms give weakly hereditary filtrations

Let $\mathbf U$ be the quantum group and let $\mathbb P=P\otimes_R\mathcal F$ be as above, where $\mathcal F$ is a finite-dimensional $\mathbf U$-module. Let $\chi=\chi_{\beta,\phi_P}$ be an induced invariant form on $\mathbb P$. A form is even when $\chi_{\beta,\phi}^\sharp=\chi_{\beta,\phi}$ and odd when $\chi_{\beta,\phi}^\sharp=-\chi_{\beta,\phi}$. A form gives a weakly hereditary filtration when, for some $h$ and for all $i$, $\mathbb A_{i,\le h}\cap\mathbb B=\mathbb B_{i,\le h}$. Even-or-odd filtration conjecture. If $\chi$ is either even or odd, then $\chi$ gives a weakly hereditary filtration. This conjecture concerns the compatibility between Weyl-group symmetry of invariant forms and the hereditary filtrations used in the representation theory of quantum groups over commutative rings. The supplied text gives no evidence of a resolution.