A linear bounded operator $T:X\to Y$ between Banach spaces is called absolutely summing if for every unconditionally convergent series $\sum_{i\in\omega}x_i$ in $X$ the series $\sum_{i\in\omega}\|T(x_i)\|$ converges.
By a famous result of Grothendieck, every operator from $\ell_1$ to $\ell_2$ is absolutely summing.
I am interested in the absolute summability of the operator of coordinatewise multiplication $$M_z:\ell_p\to\ell_{p'},\quad M_z:x\mapsto zx,$$where $p,p'\in(1,\infty)$, $\frac1p+\frac1{p'}=1$ and $z\in\ell_{p'}$. The absolute summability of the identity operator $\ell_1\to\ell_2$ implies that for every $p\in[1,2]$ and $z\in\ell_{p'}$ the multiplication operator $M_z:\ell_p\to\ell_{p'}$ is absolutely summing.
Problem. Let $p\in(2,\infty)$ and $z\in\ell_{p'}$. Is the multiplication operator $M_z:\ell_p\to\ell_{p'}$ absolutely summing?
