Super Hyperfixation Time

Details about the (alleged) rider were as follows:

Olympus' primary color was red

The dedicated rider machine for Olympus was Acrobatter, carried over from RX, but given a new red color scheme and Olympus'

Olympus would be the first rider in this history of the Kamen Rider franchise to not be a cyborg (see below for more details); however he would not have any form changes

Kotaro Minami would regain his ability to transform into Kamen Rider Black, with RX and its derivative forms being relegated to form changes

With Olympus having inherited the Acrobatter, Kotaro would have returned to using the Road Sector as his dedicated rider machine

Acrobatter (and possibly Ridoron) would have acquired human(oid) forms (either as anthropomorphization or Transformers-style)

Between RX and Olympus, the former was set to be the stronger of the two

Details stated from the (alleged) premise were:

Series creator Shotaro Ishinomori supposedly wanted Kamen Rider Olympus to be a tokusatsu version of his representative work Cyborg 009

Overall, the tone of the show was meant to be serious overall, similar to Kamen Rider Black, depicting Kotaro's guilt of having essentially destoryed Kaima World and its 5 billion residents upon defeating Emperor Crisis (クライシス皇帝) while mentoring Olympus; this would run parallel Olympus' internal reservations of having to confront his father's tyrannical ambitions (more details on the latter below)

The show's story was meant to expand upon the parallel dimension concept introduced with Kaima World (怪魔界) in Black RX with the reveal of two additional dimensions: "Olympia" (オリンビア), the home dimension of Kamen Rider Olympus, and the "World Where the Creator of Everything Resides" (すべてを創生したものがいる世界)

Each of the four worlds have their own Creation King/God (創世王); in the world of Black (RX), Gorgom chooses two candidates to become the succeeding Creation King, Emperor Crisis replicated himself endlessly, and the residents of Olympia were portrayed as ordinary citizens; in other words, Kamen Rider Olympus would have been born with the power of God

In this case, Kotaro would have become the Earth's Creation King after the collapse of Gorgom and the events of Black RX, but would have no intention of ruling, and Olympus' father would be the Creation King of Olympia

The story itself would have taken place immediately after the events of Black RX, where the Creation King of Olympia, sets his sights on conquering and ruling Earth; naturally, his son, Kamen Rider Olympus opposes this and makes contact with Kotaro

Regarding this, Kotaro was to have said the following to Olympus (rough translation; original lines are in parentheses): "If you truly love your father, you must stop him." (本当に父が好きなのなら、君は父さんをとめねばならない) "If your people were to invade this world, I won't hesitate to destroy your world." (もし君達がこの世界を侵略するのなら、僕は迷わず君の世界を滅ぼす)

"If you think I'm a devil, so be it—but I've been fighting for the justice I believe in." (僕を悪魔と思うならそれでいい、でも僕は僕の信じる正義の為に戦ってきた) "But no matter what the issue is, nothing good comes from fighting. Truthfully, I've lost everything, and here I am, all alone." (しかし、例えどんなことでも争う事からは何も生まれない、事実、僕はこうして孤独だ) "You must not let the tragedies of this world or the Kaima World repeat themselves—if you love your world, that is." (君はこの世界や、怪魔界の悲劇を繰り返してはいけない、世界を愛するならね) Thus Olympus' battle to dissuade his father and stop his ambitions begins.

Earliest known mention of the Kamen Rider Olympus rumor: - https://web.archive.org/web/20071012124712/http://salami.2ch.net/sfx/kako/988/988429642.html Japanese wiki dedicated to archiving 2channel posts dedicated to the Olympus rumor:

Mentions of the Kamen Rider Olympus rumor on Pixiv Encyclopedia:

Japanese otaku culture wiki analyzing Kamen Rider Olympus rumor:

Japanese blog mentioning Kamen Rider Olympus rumor and several other unrealized Kamen Rider projects: