Bottom-up Mergesort vs Natural Merge Sort

Natural Mergesort usually refers to Timsort. In terms of comparison to bottom-up mergesort, Timsort has $$O(n)$$ best-case running time where as bottom-up mergesort is at least $$O(n\log n)$$ on any input.
However, analysis of the worst-case running time of Timsort proves to be more tricky; an 2018 paper [1] establishes that its worst-case time complexity is $$O(n\log \rho)$$, where $$\rho$$ is the number of maximal monotonic sequences in the input array.