In Introduction to algorithm (CLRS), even in its third edition (published in 2009) it is noted in Sec 12.4 that little is known about height of randomly built binary search tree using insert and delete.
In a discussion (Oct 12. Can it be said old?) in stack ex here https://cs.stackexchange.com/questions/6342/proof-that-a-randomly-built-binary-search-tree-has-logarithmic-height as well the explanation of CLRS method has been discussed.
Is there some work in last one year or so? Much appreciate if a reference could be found of some recent publication noting the above i.e. there is no progress as yet!