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)

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)