The critical-rank vanishing conjecture for Borel regulator classes

Let $k$ be an integer with $k\geq 1$. The Borel regulator class $\beta_{2k+1}$ is a cohomology class in $H^{4k+1}(\mathrm{GL}(2k+2,\mathbb{Z});\mathbb{R})$. Critical-rank vanishing conjecture. For every integer $k=1,2,\ldots$,

$$\beta_{2k+1}=0\in H^{4k+1}(\mathrm{GL}(2k+2,\mathbb{Z});\mathbb{R}).$$

The cases $k=1$ and $k=2$ are known, while the general case remains open and determines whether $2k+2$ is the critical rank at which the corresponding Borel class vanishes.