Isomorphic $\ell^p$-subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight $w$ and an Orlicz function $\varphi$ satisfying the $\Delta_2$-condition at zero, we show that the Orlicz-Lorentz sequence space $d(w,\varphi)$ contains an $(1+\epsilon)$-isomorphic copy of $\ell_p,\, 1\le p<\infty$, if and only if the Orlicz sequence space $\ell_\varphi$ does, that is, if $p\in [\alpha_\varphi, \beta_\varphi]$, where $\alpha_\varphi$ and $\beta _\varphi$ are the Matuszewska-Orlicz lower and upper indices of $\varphi$, respectively. If $\varphi$ does not satisfy the $\Delta_2$-condition, then a similar result holds true for order continuous subspaces $d_0(w,\varphi)$ and $h_\varphi$ of $d(w,\varphi)$ and $\ell_\varphi$, respectively.