
I hope we do find out when/why Akitora ended things with Rogi. We know how obsessed Rogi is, and how he finds comfort in being part of Akitora's life, but none of the more objective details. It's interesting that despite Akitora deciding to stop communication, that he still granted Rogi his wish to be part of the Ooyamato family, by joining presumably this branch of the group only 3 years ago. Their contact must have stopped long before this, yet Akitora still formalised Rogi's place. Ane-san's death has still being left unexplained also, maybe this was the motivation for Akitora to distance himself from his son and Rogi. Reicha's parentage and the reasons why she was adopted by Rogi are still unanswered too... yet still no updates or closure on any of this.... yet! :)

It was Tatsuyuki that accepted Rogi into the group and performed the cup ceremony with him. We don't know how Akitora felt about it (though we do know Asoda was none too pleased.) What is interesting is how back in Yondaime Ooyamato Tatsuyuki, Akitora was so afraid of being indebted to Rogi that he basically sacrificed Tatsuyuki, knowing exactly what kind of person Rogi is. And now we know that Akitora was very much into Rogi back in the day. It wasn't a one-sided obsession. So whatever happened--and I do think it involves Ane-san's death--to make Akitora push Rogi away, I'm betting it will have to do with guilt on Akitora's part.

That makes sense, I made an error then thinking it was Akitora who exchanged Sake, Tasuyuki did always seem to be more accepting of Rogi, so I can see him trying to appease him by formalising his membership into the 'family", maybe he thought this would help to end his obsession. As you say, Akitora was in a relationship with Rogi, he cheated on his wife for this, and all of the characters in this series, be it Ane-san, Asoda, Akitora and Rogi have expressed 'jealousy" at one point or another. I hope we do eventually find out more about Ane-san's death, and why Akitora decided to stop communication with Rogi, Akitora it seems, also stopped being a parent to Tatsuyuki, with Asoda being the one to pick up the pieces from what it seems. Ane-san died when her son was 7, but Rogi is still hanging around Asoda & Tatsyuki at this point, it's unclear if Akitora broke things off at this point, when Tatsuyuki was 24, he says he has forgotten who Rogi is, so at some point Rogi stopped hanging around Asoda, Akitora and Tatsuyuki and probably adopted his own daughter a few years after this. We still no nothing if how Reika came into his life either! I wonder how Akitora is doing now? Maybe he has moved up to a higher position within their yakuza group, like being promoted to the executive branch, he is a widow now, who rarely sees his son, and the person who he had an affair with for nearly 3 decades still is pining for him, how does Akitora even feel about this, hopefully there will be some closure for us as readers.

I don't think Tatsuyuki did it to appease Rogi so much as it is just part of Tatsuyuki's nature to find value in everyone. Because he could not think of Rogi as disposable in spite of what he did to Tatsuyuki, he had to bring him into the group to keep him from being killed (for what he did to Tatsuyuki.)
I don't think Akitora is doing particularly well, at least financially, or he wouldn't have been so worried about being in debt to Rogi.
I do love how the title has turned out to be about those four characters and not just Rogi! And I do think we'll get closure eventually. Sensei's other stories wrap up nicely. I don't think she'll leave us hanging with this one.
latest art by sensei
https://twitter.com/scaberix/status/1022133960766177280