Sorry for asking so many questions.
I want to make sure you know the following.
Is this correct?
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Toolset has the following specification
* Toolset will give a 404 HTTP status after the second page of search results.
* Toolset uses 'index.php' after the second page
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I don't know why.
The problem occurred when the search keyword was English. It happened when the search keyword was Japanese.
This is due to theme's fault as theme uses the empty index.php as you shared details before:
It seems that the plugin you are using has a specification that the second and later pages of search results get a 404 HTTP status & that index.php is used after the second page.
The theme has an empty index.php, so it tries to use index.php and as a result nothing is displayed.
This is not Toolset issue but theme specific issue. Theme support also shared the fix and you should keep using it.