Studying the code reveals that the indentation mark must be repeated after
each blank line.  (It can also be repeated on every line, but must at a minimum
be repeated after blank lines.)  That is apparently how Markdown is suppose
to work.

Example:

~~~~~
> ~~~
First line
  Second line

>     Third line
~~~
~~~~~

The extra "`>`" at the start of "Third line" makes it work:

> ~~~
First line
  Second line

>     Third line
~~~