If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with respect to $p$-local Brown-Peterson theory $BP_*$?

[As an example, Chris Lloyd and I know that the hypothesis holds for the real Grassmanians $Gr_2(\mathbb R^m)$ with $m$ even (and $p=2$).]

If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with respect to $p$-local Brown-Peterson theory $BP_*$?

[As an example, Chris Lloyd and I know that the hypothesis holds for the real Grassmanians $Gr_2(\mathbb R^m)$ with $m$ even (and $p=2$).]

Added later: Chris and I have been having fun studying the Morava $K$-theory of $Gr_d(\mathbb R^m)$ (at $p=2$), and among other things, it seems that when $m$ is even, all of the spaces $Gr_d(\mathbb R^m)$ are $k(n)$-orientable for all $n$ and $d$. So I was just idly pondering what this means.

If a compact manifold $M$ with empty boundary is oriented with respect to all the Morava $K$-theories $K(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with respect to $p$-local Brown-Peterson theory $BP_*$?

If a compact manifold $M$ with empty boundary is oriented with respect to all the connective Morava $K$-theories $k(n)_*$, localized at a prime $p$, can one conclude that $M$ is orientable with respect to $p$-local Brown-Peterson theory $BP_*$?

[As an example, Chris Lloyd and I know that the hypothesis holds for the real Grassmanians $Gr_2(\mathbb R^m)$ with $m$ even (and $p=2$).]