  We show that there is "no stable free field of index $\alpha\in (1,2)$", in
the following sense. It was proved in a previous work by the authors, that
subject to a \emph{fourth moment assumption}, any random generalised function
on a domain $D$ of the plane, satisfying conformal invariance and a natural
domain Markov property, must be a constant multiple of the Gaussian free field.
In this article we show that the existence of $(1+\varepsilon)$-moments is
sufficient for the same conclusion. A key idea is a new way of exploring the
field, where (instead of looking at the more standard circle averages) we start
from the boundary and discover averages of the field with respect to a certain
"hitting density" of It\^o excursions.
