Is eye movement important with eyes-closed EMDR therapy? Original EMDR therapy involved a subject with open eyes tracking a target that moves side to side.

It is now common for EMDR to use devices like electronic "tappers" to alternately stimulate the left and right sides of the subject's body, leaving the subject free to close his eyes and engage in visualizations related to the trauma being treated.

In general, during wakeful visualizations with my eyes closed I often move my eyes in a manner similar to REM sleep.  I have tried a few sessions of EMDR and found it to be uncannily effective.  During closed-eye EMDR I sometimes find myself engaging in REM based on my visualizations, and other times find my eyes moving side to side in synchrony with the "tappers."

My question: Are there any studies or theories that suggest eye movement in closed-eye EMDR is relevant to the efficacy of the therapy?  E.g., does it matter whether the closed eyes move side to side if there is some other bilateral stimulation?
The state of the art in EMDR research (if not yet in practice) supports a "working memory" (WM) explanation of the therapy that does not support earlier "bilateral stimulation" hypotheses of the method.  A particularly salient summary from M.A. van den Hout et al. in Behaviour Research and Therapy 49 (2011) p.93:


  Traditionally, during EMDR, eyes are moved horizontally. In line with
  the WM account, but in contrast to original explanations, moving eyes
  vertically is equally effective (Gunter &amp; Bodner, 2008). Crucially,
  the same effects occur if WM is taxed during recall with
  non-eye-movement secondary tasks, like listening to a series of
  non-words (auditory shadowing; Gunter &amp; Bodner, 2008), drawing a
  complex figure (Gunter &amp; Bodner, 2008), or counting (Engelhard, van
  den Hout, &amp; Smeets, 2011; van den Hout et al., 2010; Kemps &amp; Tiggemann,
  2007). Tasks that are presumably hardly taxing, like simple finger
  tapping, do not have beneficial effects (van den Hout et al., 2001),
  while more complex tapping does (Andrade et al., 1997). Likewise,
  activating memories about a previously seen trauma film while playing
  taxing computer game reduced flashbacks in the week afterwards
  (Holmes, James, Coode-Bate, &amp; Deeprose, 2009).


This was backed up by correspondence with one of the authors who confirmed that in all their research protocols going back to at least 2000 research subjects must keep their eyes open so the researchers can confirm compliance.  They have no evidence this reduces the effectiveness.  In fact, the WM model suggests this should improve the effectiveness of the therapy vs. allowing subjects to close their eyes.