A new obstruction for normal spanning trees

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality.
Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833