Bounds and equality characterization for the affine type D distance

Let $w\in\widetilde{S}^D_n$ be an affine signed permutation, and let \(w)$and$\operatorname{dis}(w)$be the quantities defined in the paper. **The affine type D conjecture.** For all$w\in\widetilde{S}^D_n$,

$$\frac{\operatorname{dis}(w)}{2}\leq\$(w)\leq\operatorname{dis}(w).$$

Moreover, equality \(w)=\operatorname{dis}(w)$holds if and only if, for some$i\in[n]$and$k\in\mathbb{Z}$,

$$w=[1,\ldots,i-1,i+2k\cdot(2n+2),i+1,\ldots,n].$$

The paper presents this as an empirical conjecture for affine type $\widetilde{D}$, where the difference \(w)-\operatorname{dis}(w)/2$is suggested not to remain bounded as$w$ varies.