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：关于每个棱柱体是否存在无重叠的边展开（即长期未解的“丢勒问题”的一个特例），目前尚属未知。近年来，嵌套棱柱体的情形已被解决[Rad24]，其方法混合了两种自然展开方式：花瓣展开和带状展开。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在何种条件下确实能产生无重叠展开。特别地，我们证明了该反例在某种意义上可能是唯一可能存在的反例。尽管这一结果并未扩展已知存在边展开的形状类别，但其证明从多个层面深化了我们的理解，并开发了可能有助于解决非嵌套情形问题的工具。