For the record, a precise disproof of Lemma 3.4 seems to be in this preprint of Amar Hadzihasanovic https://arxiv.org/abs/2007.14505. Further recent developments involving diagrammatic sets are gathered in the book https://arxiv.org/abs/2404.07273 and the preprint https://arxiv.org/abs/2407.06285.

For the record, a precise disproof of Lemma 3.4 seems to be in this preprint of Amar Hadzihasanovic Diagrammatic sets and rewriting in weak higher categories. Further recent developments involving diagrammatic sets are gathered in the book Combinatorics of higher-categorical diagrams by the same author, and the preprint Diagrammatic sets as a model of homotopy types by Chanavat and Hadzihasanovic.