Twisted-center unit conjecture for Clifford algebras

Let $A$ be a commutative ring with unit in which $2\neq0$, let $(N,Q)$ be a quadratic module over $A$, and set $\mathcal{C}:=\mathcal{C}(N,Q)$. Let $e$ be the idempotent used to form the Clifford algebra $e\mathcal{C}$, and write $\widetilde{Z}(e\mathcal{C})^{\times}$ for its group of units in the twisted center, with decomposition into $+$ and $-$ parts denoted by $\alpha=\alpha_{+}+\alpha_{-}$; write $\mathcal{C}_{+}[2]$ for the elements of the even component annihilated by $2$. Twisted-center unit conjecture. The intersection

$$\widetilde{Z}(e\mathcal{C})^{\times}\cap\bigl(\mathcal{C}_{+}[2]\oplus\mathcal{C}_{-}\bigr)$$

is empty; equivalently, no element $\alpha\in\widetilde{Z}(e\mathcal{C})^{\times}$ satisfies $2\alpha_{+}=0$. The conjecture is presented as a proposed relaxation of the conditions in the preceding proposition; its general validity is left open.