My question seems to be easy but I cannot spot the answer. I am interested in ranges of operators defined on $c_0$. The celebrated "operator version" of Sobczyk's theorem says that if we are given a separable Banach space $X$ and its subspace $Y$, then every bounded operator $T\colon Y\to c_0$ can be extended to a bounded operator $\overline{T}$ with $\|\overline{T}\|\leq 2\|T\|$ (categorically speaking, $c_0$ is "separably injective"). I am wondering if I could use this theorem (or anything else) to (dis)prove the following conjecture:
If $X$ is a $c_0$-saturated separable Banach, then the range of every operator $T\colon c_0\to X$ embeds into $c_0$. We know that (consult Lindenstrauss and Tzafriri's book) every quotient of $c_0$ embeds into $c_0$ but how about ranges of this sort of operators?