Self-similar solution-count conjecture for the nonlinear parabolic system

Consider the boundary-value problem for a function $w$ on $[0,\infty)$:

$$w” + \frac{n+1}{r} w' -\kappa r w' + 3 r w w' + (n+2) w^2 - \kappa w =0,$$

with

$$w(0)=\alpha>0,\qquad w'(0)=0,\qquad w(r)\sim r^{-2}\quad\text{as }r\to\infty,$$

where $\kappa>0$ and $n>4$. Define

$$\nu(n)=\min\left\{k\in\mathbb{N}\mid k\geq\frac{n+2}{n-4}\right\}.$$

Self-similar solution-count conjecture. For fixed $\kappa>0$ and $n>4$, this boundary-value problem has $\nu(n)-2$ non-trivial solutions.

These solutions describe radial self-similar singularities of the parabolic system. The conjectured count is supported by analytical considerations and numerical computations, while existence of all the asserted solutions remains open in the source.