Given a constant $q\in (1,\mathrm{\infty })$, 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 ${\mathrm{\ell }}^q$ 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 ${\sum }_{n=1}^N{\mid x_n^{\ast }\mid }^q\le {\sum }_{n=1}^N{\mid y_n^{\ast }\mid }^q$ 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 ${\mathrm{\ell }}^1$ and ${\mathrm{\ell }}^q$. More specificially, we first show that every space which is a normed interpolation space with respect to the couple $({\mathrm{\ell }}^p,{\mathrm{\ell }}^q)$ 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 ${\mathrm{\ell }}^q$, then it is a normed interpolation space with respect to the couple $({\mathrm{\ell }}^1,{\mathrm{\ell }}^q)$. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in arXiv:1703.04254v1 [math.OA].