It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.

翻译：目前尚不清楚是否每个棱柱体都存在非重叠的边展开，这是长期未解决的"丢勒问题"的一个特例。最近，嵌套棱柱体的情况已通过混合（在某种意义上）两种自然展开方式——花瓣展开与带状展开——得到解决[Rad24]。带状展开因一个特定反例[O'R13b]而失效。本文的主要贡献在于刻画了嵌套棱柱体的带状展开何时确实能产生非重叠展开。特别地，我们证明了上述反例在某种意义上具有唯一可能性。尽管这一结果并未扩展已知存在边展开的几何体类别，但其证明过程通过发展可能有助于解决非嵌套情况的分析工具，从多个维度深化了我们的理论认知。