  Given a constant $q\in(1,\infty)$, we study the following property of a
normed sequence space $E$:
  =====================
  If $\left\{ x_{n}\right\}_{n\in\mathbb{N}}$ is an element of $E$ and if
$\left\{ y_{n}\right\}_{n\in\mathbb{N}}$ is an element of $\ell^{q}$ such that
$\sum_{n=1}^{\infty}\left|x_{n}\right|^{q}=\sum_{n=1}^\infty
\left|y_{n}\right|^{q}$ and if the nonincreasing rearrangements of these two
sequences satisfy
$\sum_{n=1}^{N}\left|x_{n}^{*}\right|^{q}\le\sum_{n=1}^{N}\left|y_{n}^{*}\right|^{q}$
for all $N\in\mathbb{N}$, then $\left\{ y_{n}\right\}_{n\in\mathbb{N}}\in E$
and $\left\Vert \left\{ y_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}\le
C\left\Vert \left\{ x_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}$ for some
constant $C$ which depends only on $E$.
  =====================
  We show that this property is very close to characterizing the normed
interpolation spaces between $\ell^{1}$ and $\ell^{q}$. More specificially, we
first show that every space which is a normed interpolation space with respect
to the couple $\left(\ell^{p},\ell^{q}\right)$ for some $p\in[1,q]$ has the
above mentioned property. Then we show, conversely, that if $E$ has the above
mentioned property, and also has the Fatou property, and is contained in
$\ell^{q}$, then it is a normed interpolation space with respect to the couple
$\left(\ell^{1},\ell^{q}\right)$. These results are our response to a
conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in
arXiv:1703.04254v1 [math.OA].
