Add Box to multiline-expression which is part of aligned-equation?

I want to add a box (black color) to encapsulate the last two lines of this triple-line-aligned expression.

As you can see, the current solution inspired by this post is not perfect because it misses some topmost components.

Note that the target is a part of the aligned-expression so the general commends like boxed or empheq may not work.

newcommand{pstrut}[1]{vrule height0pt depth0pt width0pt #1 fboxsep}
newcommand*bmarkeq{stepcounter{markeq}%
  tikz[remember picture]node(startframe-themarkeq){pstrut{height}};%
  kernfboxsep}
newcommand*emarkeq{kernfboxsep
  begin{tikzpicture}[remember picture,overlay]
    node (endframe-themarkeq){pstrut{depth}};
    draw[,black,opacity=0.8] (startframe-themarkeq.north) 
      rectangle (endframe-themarkeq.south);
  end{tikzpicture}%
}


begin{align*}
    mathrm{L}_{c}(theta) 
    = &sum_{i=1}^{m} ln pleft(x_{i}, z_{i} ; thetaright) 
    = bmarkeq &sum_{i=1}^{m}left(z_{i}left[ln (pi)+x_{i} ln left(p_{r}right)+left(1-x_{i}right) ln left(1-p_{r}right)right] 
    &+ left(1-z_{i}right)left[ln (1-pi)+x_{i} ln left(p_{b}right)+left(1-x_{i}right) ln left(1-p_{b}right)right]right) emarkeq
end{align*}

documentclass{article}
usepackage{mathtools}

begin{document}

begin{align*}
    mathrm{L}_{c}(theta)
    &=sum_{i=1}^{m} ln p(x_{i}, z_{i} ; theta) 
   & =boxed{begin{aligned}[t] &sum_{i=1}^{m}bigl(z_{i}bigl[ln (pi)+x_{i} ln (p_{r})+(1-x_{i}) ln(1-p_{r})bigr]
   &+(1-z_{i})bigl[ln (1-pi)+x_{i} ln(p_{b})+(1-x_{i}) ln(1-p_{b})bigr]bigr)end{aligned}}
end{align*}

begin{align*}
    mathrm{L}_{c}(theta)
    &=sum_{i=1}^{m} ln p(x_{i}, z_{i} ; theta) 
Aboxed{& ={begin{aligned}[t]&sum_{i=1}^{m}bigl(z_{i}bigl[ln (pi)+x_{i} ln (p_{r})+(1-x_{i}) ln(1-p_{r})bigr]
   &+(1-z_{i})bigl[ln (1-pi)+x_{i} ln(p_{b})+(1-x_{i}) ln(1-p_{b})bigr]bigr)end{aligned}}}
end{align*}

end{document}