Suslin's conjecture on universal vanishing of $\operatorname{SK}_1$

Let $F$ be a field, let $A$ be a central simple $F$-algebra, and for every extension $L/F$ write $\operatorname{SK}_1(A\otimes_F L)$ for the reduced Whitehead group. Suslin's conjecture. If

$$\operatorname{SK}_1(A\otimes_F L)=0$$

for every extension $L/F$, then the index of $A$ is square-free. Merkurjev proved the assertion whenever $4$ divides the index, but the general conjecture remains open.