\version "2.22"

spanTest = {
    \override TextSpanner.bound-details.right-broken.text = ##f
    \override TextSpanner.bound-details.right.text = "R"

    \override TextSpanner.bound-details.left.text = "L"
    \override TextSpanner.bound-details.left-broken.text = \markup\right-align "(L)"

    <>\startTextSpan
    s1
    \break
    <>\stopTextSpan
    s1
    <>\startTextSpan
    s1
    <>\stopTextSpan
  }

\new Staff {
  \spanTest
}

\new Staff {
  \override TextSpanner.to-barline = ##t
  % removes spurious "(L)__R", but fails to create "R"

  \spanTest

  s4 % makes last "right" appear (but extends piece)
}