Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
No, the conjecture is still wide open for rank $r\geq 2$.
The closest thing to progress is the work of Bhargava and Shankar that quantifies the rank $0$ case and shows that BSD holds for a positive proportion of elliptic curves. You can find it here:
- Manjul Bhargava & Arul Shankar, "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0" (2015)
For subsequent developments, see for example
- Manjul Bhargava, Christopher Skinner & Wei Zhang, "A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture" (2014, preprint)
Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned is the work of Yun and Zhang. It is about the function field analogue but they obtain information about the full Taylor series of the $L$-function. The paper has recently been accepted at Annals of Mathematics.