Embedding video in multiple-slide-per-page presentation

Question

I am preparing a presentation for an interactive session, and need all of my four slides to be shown at once. In order to do this, I use handout mode (and don't have any slide transitions). In one of the slides, I have a video. I run my presentation with pdfpc.

While the video shows up as expected with the handout option disabled, it shifts when I turn it on. I understand from this answer that this is due to hyperlinks do not get scaled properly when producing multiple frames per page.

However, I would like to produce exactly that: a multiple-frame-per-page presentation embedding a video in one of the frames. Any suggestions how I could achieve that?

Here is my MWE, but I suspect the solution (if one exists) might be something quite different not using handout at all.

\documentclass[handout]{beamer}

\usepackage{units}
\usepackage{color}
\usepackage{multimedia}
\mode<presentation>
{
\usetheme{Madrid}
\usecolortheme{beaver}
\setbeamercovered{invisible}
}

\usepackage{xcolor}
\definecolor{RawSienna}{cmyk}{0,0.72,1,0.45}

\usepackage{hyperref}

\usepackage[english]{babel}

\usepackage{mathptmx}
\usepackage[scaled=.90]{helvet}
\usepackage{courier}

\title[]{\emph{Presentation}}

\author{Me}

\institute[Group]{
University
}

\usepackage{pgf,pgfpages}
\pgfpagesuselayout{4 on 1}[a4paper,border shrink=5mm,landscape]

\begin{document}

\begin{frame}
\frametitle{First frame}
  \begin{itemize}
  \item First frame
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Second frame}
  \begin{itemize}
  \item Second frame
  \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Third frame}
  \begin{overlayarea}{\textwidth}{0.37\textheight}
\centering
\vspace{0.05cm}
\begin{minipage}{.44\textwidth}
  \centering
  \includegraphics[width = 0.6\textwidth]{static_image.png}\\
  Image OK
\end{minipage}
\begin{minipage}{.44\textwidth}
  \centering
    \href{run:video.avi?autostart&loop}{\includegraphics[width=0.6\textwidth]{placeholder_image.png}}\\%
  Movie - shifted     
\end{minipage}
  \end{overlayarea}
  \begin{overlayarea}{\textwidth}{0.51\textheight}
\begin{itemize}
  \item Third, problem frame
\end{itemize}
  \end{overlayarea}
\end{frame}

\begin{frame}
  \frametitle{Fourth frame}
  \begin{itemize}
  \item Fourth frame
  \end{itemize}
\end{frame}  

\end{document}