The question is only about elliptic curves $E$ over $\mathbb{Q}$ and concerns only the aspect 

(order of vanishing of $L(E,s)$ at $s=1$)$\ =\ $(rank of $E(\mathbb{Q})$).

Let $r$ be the LHS and $d$ the RHS, so that (a special case of ) the Birch and Swinnerton-Dyer Conjecture is

**BSD?.** $r=d$.

By the end of the last millenium, we knew

**Theorem** (1977--2000). *If* $\ r=0,1$, *then* $d=r$ (*and* $\ \operatorname{Sha}(E)$ *is finite*).

Some years ago, I heard that there was some progress in proving $(r>0)\Longrightarrow (d>0)$ under the assumption of the finiteness of $\operatorname{Sha}(E)$.  What is the current status of the

**Statement.**  *Suppose that* $\operatorname{Sha}(E)$ *is finite.  If* $r>1$, *then* $d>0$ ?