  We continue the study of local $Tb$ theorems for square functions defined in
the upper half-space $(\mathbb{R}^{n+1}_+, \mu \times dt/t)$. Here $\mu$ is
allowed to be a non-homogeneous measure in $\mathbb{R}^n$. In this paper we
prove a boundedness result assuming local $L^q$ type testing conditions in the
difficult range $q \in (1,2)$. Our theorem is a non-homogeneous version of a
result of S. Hofmann valid for the Lebesgue measure. It is also an extension of
the recent results of M. Lacey and the first named author where non-homogeneous
local $L^2$ testing conditions have been considered.
