On the relationship between the factorizations of an operator $T$ and its second adjoint $T^{**}$

Question

Let $T$ be an operator from a space $X$ into a space $Y$ and let $1\leq p<\infty$. If $T^{**}$ has a factorization $T^{**}=RS:X^{**}\xrightarrow{S} l_{p}\xrightarrow{R}Y^{**}$, where $S$ is compact and $R$ is an operator. Does $T$ has a factorization $T=AB:X\xrightarrow{B} l_{p}\xrightarrow{A}Y$, where $B$ is also compact and $A$ is an operator? Of course, this question is true if $Y$ is a dual space, because then $Y$ is norm one complemented in $Y^{**}$.