First lets fix a prime $p$ (I really care about $p=2$ but would be happy to know about other primes as well). When localized at a prime the spectrum $ku$ (Complex connective K-theory) splits as a wedge of suspensions of a spectrum $l$. In their paper on the Cooperation algebra of the Adams Summand, Baker-Richter state that the complex orientation $MU \to l$ factors through the complex orientation of $BP$ and that it can be arranged so that the image of $x_n \in \pi_{2n}MU$ and $v_i \in \pi_{2p^i-2}BP$ go to zero in $\pi_*l$. Let $BP(n)$ denote the "truncation" of $BP$ so that $\pi_*BP(n) \cong \pi_*BP/(v_{n+1},v_{n+2},...)$. (This may not be the notation in the literature, but I am pretty sure I have seen people mention these objects).

My question is can we extend the complex orientation of $l$ over some of these other spectra? Does the structure of $l$ and $l_*BP$ as a module over $BP(n)$ change a lot when going from $BP(n)$ to $BP(n+1)$?