  We prove a local $Tb$ theorem under close to minimal (up to certain
`buffering') integrability assumptions, conjectured by S. Hofmann (El Escorial,
2008): Every cube is assumed to support two non-degenerate functions $b^1_Q\in
L^p$ and $b^2_Q\in L^q$ such that $1_{2Q}Tb^1_Q\in L^{q'}$ and
$1_{2Q}T^*b^2_Q\in L^{p'}$, with appropriate uniformity and scaling of the
norms. This is sufficient for the $L^2$-boundedness of the Calderon-Zygmund
operator $T$, for any $p,q\in(1,\infty)$, a result previously unknown for
simultaneously small values of $p$ and $q$. We obtain this as a corollary of a
local $Tb$ theorem for the maximal truncations $T_{\#}$ and $(T^*)_{\#}$: for
the $L^2$-boundedness of $T$, it suffices that $1_Q T_{\#}b^1_Q$ and $1_Q
(T^*)_{\#}b^2_Q$ be uniformly in $L^0$. The proof builds on the technique of
suppressed operators from the quantitative Vitushkin conjecture due to
Nazarov-Treil-Volberg.
