This is not a full solution, but some observations that may help get you there.

All the moving pieces still have an orange or red facelet. Therefore red is opposite orange. If you place the four orange corners relative to each other, they only go one way and therefore determine the colour scheme and hence the corners colours, too. We can also put in the centres according to the colour scheme (it does not matter which orientation - there can be no parity problem due to the identical edges) and that results in the following:

         R G w
         R o O
         g O g
 y . W   R . R   Y . O   B . B
 O b W   . w .   R g .   O y .
 Y . Y   g G O   G . b   W R b
         O r w
         R . B
         O . R

You can then try to solve it without putting any further colours in. It should be possible to do this, though I have not checked. A corners first solution method is very useful for this.

This is not a full solution, but some observations that may help get you there.

All the moving pieces still have an orange or red facelet. Therefore red is opposite orange. If you place the four orange corners relative to each other, they only go one way and therefore determine the colour scheme and hence the corners colours, too. We can also put in the centres according to the colour scheme (it does not matter which orientation - there can be no parity problem due to the identical edges) and that results in the following:

         R G w
         R o O
         g O g
 y . W   R . R   Y . O   B . B
 O b W   . w .   R g .   O y .
 Y . Y   g G O   G . b   W R b
         O . w
         R r B
         O . R

You can then try to solve it without putting any further colours in. It should be possible to do this, though I have not checked. A corners first solution method is very useful for this.