"lineno" not compatible with "balance"

Question

In the file with two columns by twocolumn options, the last page can't be balanced by balance package if lineno package works.

\documentclass[twocolumn]{book}
\usepackage{lipsum,balance}
\usepackage[switch]{lineno}
\begin{document}    
\twocolumn[
\begin{@twocolumnfalse}
\chapter{the first}
\end{@twocolumnfalse}
]
\linenumbers
\lipsum[1-8]
balabala
\balance
\end{document}

produces:

Obvious, the last page is not balanced. So, how to balance the last page based on lineno working.

---

As pbalance would work in sometime, but when there are figures with figure* parameter, pbalance would be very difficult to be adjusted well.

\documentclass[twocolumn]{book}
\usepackage{lipsum,pbalance,stfloats,graphicx}
\usepackage[switch]{lineno}
%\shrinkLastPage{2.2in}
\begin{document}

\twocolumn[
\begin{@twocolumnfalse}
\chapter{the first}
\end{@twocolumnfalse}
]
\linenumbers
\lipsum[1]
\par
\begin{figure}[b]

\centering
\includegraphics[width=0.4\textwidth]{example-image-a}
\end{figure}
\lipsum[2-8]
balabala
\end{document}

produces:

Answer 1

You may want to try the pbalance package. Caveat: the author states that it is hackish and of beta quality, so it doesn't always work and sometimes the package would just give up; but it at least sort-of works on a test document.

Try 1

We begin by trying to follow the documentation and just loading pbalance, this gives us the test document

\documentclass[twocolumn]{book}
\usepackage{lipsum,pbalance}
\usepackage[switch]{lineno}
\begin{document}    
\twocolumn[
\begin{@twocolumnfalse}
\chapter{the first}
\end{@twocolumnfalse}
]
\linenumbers
\lipsum[1-8]
balabala
\end{document}

The output goes in the correct direction:

Try 2

The first try almost worked, but it looks like we can move another line from the left column to the right. According to the documentation, we can manually adjust this via \shrinkLastPage. So let's try eyeballing where the break should be. After some trial and error, I settled on 4.1in

\documentclass[twocolumn]{book}
\usepackage{lipsum,pbalance}
\usepackage[switch]{lineno}
\shrinkLastPage{4.1in}
\begin{document}

\twocolumn[
\begin{@twocolumnfalse}
\chapter{the first}
\end{@twocolumnfalse}
]
\linenumbers
\lipsum[1-8]
balabala
\end{document}

And this time it is almost completely balanced.

Additional Caveats

Since you used the book document class, I am assuming eventually you want to use this to balance the last page of each individual chapter. As it currently stands, the pbalance package cannot do that.

However, the trick that it uses can be implemented manually, and may be easier to do than using \newpage as my comment suggested. Basically: after the content of the document has stabilized, compile your document first. Then issue \enlargethispage{<len>} where you would've put \balance (in the first column of the final page), with a negative length that you determine manually to get the two columns to balance. Repeat for each chapter.