I'm not an expert in the field but the Related Work section in the paper has quite a lot of references and seem to cover significant ground. Some quotes from that section:

Recently, there has been a body of work following this structure with Taylor expansions of order higher than two [40, 9, 12, 28, 29, 25]. Unlike our paper, these works are in the setting of convex optimization, do not study the complexity of minimizing the regularized Taylor expansion in each iteration [...]

To the best of our knowledge, no efficient algorithm for higher-order Newton methods of degree d > 3 has been presented. In fact, designing such an algorithm is referred to as an open problem in [23, Sec. 1.5] and [25, Sec. 5].

To our knowledge, the only works that establish superlinear rates of local convergence for higher-order Newton methods are [47] and [24] (and the related PhD thesis [23]), the latter of which came to our attention at the time of writing this paper.

Seems highly unlikely it's anything close to the example you posted.