Current document is not properly set when calling LiveTable Results in LiveData

Reproduction steps:

• Create a dedicated LiveTable results containing some code such as:
  noformat
  velocity wiki="false"
  #set ($lastAuthor = $doc.contentAuthorReference)
  #set ($currentUser = $xcontext.userReference)
  #set ($discard = $xcontext.context.setUserReference($lastAuthor))
  #gridresultwithfilter("XWiki.XWikiUsers" $request.collist.split(","))
  #set ($discard = $xcontext.context.setUserReference($currentUser))
  /velocity
  noformat
• Create a livedata using that result page

Expected result:

• the livedata displays the entries by using the defined context user

Obtained result:

• the livedata doesn't display any entry, only N/A rows. However it works by resoluving manually the author.

The reason of the error is apparently that $doc is not properly assigned when the request is made in LiveData REST API to get LT results.