Malheureusement, this is not true, not even for the weak order.  This can be seen for example when $G = GL(4)$ and $K = GL(2) \times GL(2)$.  Then $K \backslash G / B$ is parameterized by involutions with signs attached to fixed points and the map $\varphi$ simply forgets the markings on the fixed points.

For example, the closed orbit associated to $(1^+)(2^-)(3^-)(4^+)$ lies below both $(14)(2^-)(3^+)$ and $(14)(2^+)(3^-)$.  Figures with weak order for a plethora of examples of symmetric subgroups appear in [a preprint by Ben Weyser][1].


  [1]: http://arxiv.org/abs/1201.4397