Optimality conjecture for vanishing orders of Witten zeta functions

Let $G$ be a compact simply connected Lie group identified with its root system $\Phi$, and write $\zeta_\Phi(s):=\zeta_G(s)$. For negative even integers, the theorem states

$$\underset{s \in -2\mathbb{N}}{\operatorname{ord}}(\zeta_\Phi(s))\geq \operatorname{Rank}(\Phi).$$

For negative odd integers, it states

$$\underset{s \in 1-2\mathbb{N}}{\operatorname{ord}}(\zeta_\Phi(s))\geq \begin{cases} \left\lfloor \frac{n}{2} \right\rfloor & \Phi=A_n,\\ 1 & \Phi=D_n\text{ with }n\geq 5\text{ and odd},\\ 2 & \Phi=E_6,\\ 0 & \text{otherwise}. \end{cases}$$

Optimality conjecture. In the theorem, all stated inequalities for orders of vanishing are actually equalities. The theorem establishes the displayed lower bounds for compact simply connected Lie groups; the conjecture asserts that none of these bounds can be improved.