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As all internal links from that single page become broken, it goes into perpetual mode which makes Google crawling "new" pages throughout the site which actually are all duplicated content.
It should send a 404 header if not.It does now. The whole story here is that shared hosting configured as CGI or CGI/FastCGI (as far as I experienced with multiple hosts) opens this "slash" thing if parse HTML as PHP is used.
Additionally, link to other pages with a leading slash, including the full path.That has never been an issue. Just when trailing slash appears after file extension like .html because a search engine has picked it somewhere.