The sign conjecture for Coxeter arrangements of types A and D

Let $V$ be an $n$-dimensional space, let $\mathcal{H}$ be a Coxeter arrangement of type $A_n$ or $D_n$ in $V$, and let $a\in V$ satisfy $0\in\mathbb{B}(a,R)$. Assume that $a$ lies in the interior of a chamber $T$ of $\mathcal{H}$. Sign conjecture. If $\mathcal{H}$ has type $A_n$ and $n\equiv 2$ or $3\pmod 4$, then

$$(-1)^{\lfloor (n+1)/4\rfloor}(-1)^T P(\mathcal{H},\mathbb{B}(a,R))>0.$$

If $\mathcal{H}$ has type $D_n$ and $n$ is odd, then

$$(-1)^T P(\mathcal{H},\mathbb{B}(a,R))<0.$$

This refines the parity-mismatch conjecture by predicting the sign rather than merely nonvanishing. The source presents computational evidence in low dimensions, while the general assertion remains open.