A conjecture on the behavior of tails of fixed points of the shot noise transform

We show, under some restrictions on the response function depending on a parameter $\alpha$, that the tails of fixed points of transforms of a Poisson shot noise process are proportional at infinity to the exponential function of order $-\alpha$ if $\alpha\in(1,2)$. We advance an argument in support of the conjecture that this result remains true for $\alpha\geq 2$.