Continuity of weighted estimates in $A_{p}$ norm

We prove that for a general Calderón-Zygmund operator $T$ the numbers $\Vert T\Vert _{L^{p}(w)\rightarrow L^{p}(w)}$ converge to $\Vert T\Vert _{L^{p}(dx)\rightarrow L^{p}(dx)}$ as the $A_{p}$ norm of $w$ converges to $1$, i.e. as $[w]_{A_{p}}\rightarrow 1^{+}$ for $1<p<\infty$.