Classification conjecture for finite-mass entire Keller–Segel solutions

Let $(u,v)$ be an entire solution of the Keller–Segel system on $\mathbb{R}^2\times\mathbb{R}$, meaning that $u$ and $\nabla v$ are smooth and satisfy the system, with the second equation understood through the representation formula. Assume the finite mass condition

$$\sup_{t\in\mathbb{R}}\int_{\mathbb{R}^2}u(x,t)\,dx<+\infty.$$

The solution is the finite-mass entire-solution classification conjecture. Any nontrivial entire solution satisfying the finite mass condition has the form

$$\left\{\begin{aligned} &u(x,t)\equiv \frac{8\lambda^2}{(\lambda^2+|x-\xi|^2)^2},\\ &v(x,t)\equiv -2\log\left(\lambda^2+|x-\xi|^2\right)+C, \end{aligned}\right.$$

for some $\lambda>0$, $\xi\in\mathbb{R}^2$, and $C\in\mathbb{R}$. This conjecture would classify all nontrivial finite-mass entire solutions as stationary Liouville profiles; the preceding blow-down theorem motivates it, but the time-independence and complete classification of the original entire solution remain to be established.