Cross staff stems for beamed notes

[Helge Kruse]

I need to add cross staff stems and used the approach found in
http://lilypond.org/doc/v2.19/Documentation/snippets/staff-notation.html#staff-notation-cross-staff-stems

Unfortunately this doesn't give the expected result for eighth notes.
- If the stem points down, cross staff is not working
- The beam is visible in both staves.
- If I use manual beaming to remove the beam, the eighth note is
displayed as a fourth.

How can I resolve all three problems?

Regards,
Helge

\version "2.19.81"

\score {
  <<
    \new PianoStaff <<
      \new Staff { \time 6/8 fis'4. \stemDown a' }
      \new Staff { \time 6/8 \clef bass \crossStaff { <fis fis,>4. < a!
a,!> } }
    >>
    \new PianoStaff <<
      \new Staff { \time 6/8 r8 <a' a''> q \stemUp q[ q] \stemDown q  }
      \new Staff { \time 6/8 \clef bass r8 \crossStaff { a8 a  \stemUp
a[ a] \stemDown a } }
    >>
  >>

  \layout {
    \context {
      \PianoStaff
      \consists #Span_stem_engraver
    }
  }
}