Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying \begin{equation}1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ and }\;\;\;\;\lim_{n\to\infty}w_n=0.\end{equation} We define the Rosenthal-Woo space $X_{\infty,p,w}:=[h_n]_{n=1}^\infty$ the closed linear span of vectors \begin{equation}h_n:=f_n\oplus w_ne_n\in c_0\oplus_\infty\ell_p,\end{equation} where $(e_n)_{n=1}^\infty$ and $(f_n)_{n=1}^\infty$ are the respective canonical bases of $\ell_p$ and $c_0$, and $\oplus_\infty$ denotes the direct sum under the sup norm. It was shown in [Woo75] that the spaces $X_{\infty,p,w}$ are all pairwise isomorphic for all weights $w=(w_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty w_n^p=\infty$.
Meanwhile, the Lorentz sequence space $d(w,p)$ is defined as the completion of $c_{00}$ under the norm \begin{equation}\|(a_n)_{n=1}^\infty\|_d=\sup_{\sigma\in\Pi}\|(a_{\sigma(n)}w_n^{1/p})_{n=1}^\infty\|_p,\;\;\;(a_n)_{n=1}^\infty\in c_{00},\end{equation} where $\Pi$ is the set of all permutations of $\mathbb{N}$ and $\|\cdot\|_p$ is the usual norm in $\ell_p$.
Let us denote $w^p=(w_n^p)_{n=1}^\infty$ so that $d(w^p,p)$ has norm \begin{equation}\sup_{\sigma\in\Pi}\|(a_{\sigma(n)}w_n)_{n=1}^\infty\|_p,\;\;\;(a_n)_{n=1}^\infty\in c_{00},\end{equation}
Observation (?). It's easy to see that the space $d(w^p,p)$ is the "symmetrization" of the space $X_{\infty,p,w}$. In other words, \begin{equation}\|\sum_{n=1}^\infty a_nd_n\|=\sup_{\sigma\in\Pi}\|\sum_{n=1}^\infty a_{\sigma(n)}h_n\|\end{equation} Hence (?), the spaces $d(w,p)$ are pairwise isomorphic for all weights $w\notin\ell_p$.
But this looks like it is false, as it is an extremely strong conclusion.
Question. Where, if anywhere, have I gone wrong in my argument above? I must be missing something obvious, but for the life of me I do not see it.
Or, is it possible that Woo made a mistake in his paper? After all, his proof consists of nothing more than the hand-waving sentence fragment "The same as that of Theorem 13 in [Ro70]" (referring to Rosenthal's famous paper).
Thanks!