Symmetric basis conjecture for eigenvectors of generators on $\ell_p$

Let $A$ be the generator of a $C_0$-group on $\ell_p$, with $p>1$, whose eigenvalues, counted with multiplicity, are $\{\lambda_n\}_{n=1}^{\infty}$ and whose corresponding normalized eigenvectors are $\{e_n\}_{n=1}^{\infty}$. Assume that

$$\overline{\operatorname{Lin}}\{e_n\}_{n=1}^{\infty}=\ell_p.$$

Also assume that the point spectrum $\{\lambda_n\}_{n=1}^{\infty}$ satisfies condition (2). Symmetric basis conjecture. Then $\{e_n\}_{n=1}^{\infty}$ forms a symmetric basis of $\ell_p$. This proposes an extension of the preceding result for operators with symmetric basis families of eigenvectors and would characterize the eigenvectors under the stated spectral and completeness hypotheses; no resolution is given in the source.