Nonexistence conjecture for singular self-similar profiles with intermediate exponent

Let $\gamma\in(0,1)$, and consider the singular self-similar profile equation

$$$$

Nonexistence conjecture. The equation does not have solutions.

The methods used for the endpoint cases $\gamma=0$ and $\gamma=1$ do not appear to apply when $\gamma\in(0,1)$. The conjecture is motivated by partial analytic results and numerical computations; no resolution is stated here.