Unconditional convergence for wavelet frame extensions

Let $\{\psi_{j,k}\}_{(j,k)\in\mathbb Z^2}$, $\{\widetilde\psi_{j,k}\}_{(j,k)\in\mathbb Z^2}$ be dual wavelet frames in $L_2(\mathbb R)$, let $\eta$ be an even, bounded, decreasing on $[0,\infty)$ function such that
$$ \int_0^\infty\eta(x)\ln(1+x)\,dx<\infty, $$
and $|\psi(x)|,|\widetilde\psi(x)|\le\eta(x)$. Then the series $\sum_{j,k\in\mathbb Z}(f,\widetilde\psi_{j,k})\psi_{j,k}$ converges unconditionally in $L_p(\mathbb R)$, $1<p<\infty$.