So here is the bare bones of why many believe there will be a fourth episode/"lost special" to cap off season four of BBC's Sherlock. This is in no particular order and will touch upon other theories centric to the "lost special" conspiracy.
1)The line "There must be something comforting about the number 3; people always stop counting after 3.
2) The pay-off of The Final Problem (season finale) did not match up with the hints dropped by the members of the production team before season four aired. Indeed the decline in quality of production echoes the original "Final Problem" when Sir Arthur Conan Doyle purposefully tanked a story just so he wouldn't have to keep continue with the Sherlock stories. Fans made such a fuss (inclusive of, but not limited to, physically assaulting ACD) that Doyle was forced to bring Sherlock back, starting with a lost story. In fact I believe Moftiss are intentionally trying to re-create this furor.
3) The relationship between Sherlock and Watson has always been the focus of the BBC/Moftiss adaptation, so making the (possibly) series finale all about a lost Holmes sibling doesn't fit into the established narrative arc.
4) The Final Problem also does not match the arc shared by Sherlock and Moriarty. Moriarty repeatedly stated that his ultimate goal was to burn the heart out of Sherlock. Even if you don't believe a romantic relationship exists between Sherlock and John, the series and the creators have made it explicitly clear that John Watson is Sherlock's heart.
5) The production staff keep releasing Setlock photos that don't match with the aired episodes.
6) The production team has not engaged in their, by now, typical 'victory' tour and have been incredibly tight lipped about the whole season in general.
7) Executive Producer Sue Vertue previously stated that Sherlock seasons may not always be three episodes.
8) In the past spoilers have been religiously guarded against, however, when The Final Problem leaked they not only failed to contain it, as they had successfully in the past, but they called attention to it and posting a link to the leaked content.
9) The Final Problem contradicts rules that have been previously established within the BBC Sherlock universe, i.e. being uninjured after a massive explosion, Sherlock ignoring "Vatican Cameos", and Mycroft's line from His Last Vow about there not being a prison capable of holding Sherlock Holmes to name a few.
10) The fact that Steven Moffat made an unforced error when he said there was no lost special, something he hadn't been asked about.