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 $d>0$ ?