The Wikipedia article for the quantum Fisher Information mentions that one can expand the Bures fidelity and the quantum Fisher Information will appear as the second-order correction term.
However in the cited paper (https://arxiv.org/abs/1612.04581) — and in the paper that derives the symmetric expansion — the explicit dependence on the higher order correction terms are omitted. On the Wikipedia page it says that my fidelity may be expanded as $$ F(\rho_{\theta}, \rho_{\theta + d\theta}) = 1 - \frac{1}{4} \sum_{i,j} (F_Q^{ij}(\rho(\theta)) + 2 \sum_{\lambda_k(\theta) = 0} \partial_i \partial_j \lambda_k) d\theta_i d\theta_j + O(d\theta^4). $$
My question now is, how can I derive that $O(d\theta^4)$ term? If I follow the calculations from the reference, I would only know that the term is $O(d\theta^3)$. The same has been done in this question: How is the connection between Bures fidelity and quantum Fisher information derived?
Is the formula on Wikipedia incorrect or is there some way to see that the remainder is actually of order $O(d\theta^4)$?
