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It was expected that Hakuho would get a position among the elders for life solely on his yokozuna record (like previously Taiho, Kitanoumi and Takanohana – otherwise yokozuna get temporary 5 year and ozeki 3 year position without elder stock) – but the yokozuna committee recently suggested the practice should be discontinued, which was a bit controversial since it came just in time for Hakuho's expected retirement. In practice this doesn't mean much (Hakuho already called dibs on Magaki elder stock), except that he will not be able to open a Hakuho stable.
Now, the current Miyagino-oyakata, where Hakuho trained, is 64 himself so he's expected to retire soon, perhaps already next year (cap is 65, but can be extended to 70 if approved by a board). So the most likely scenario is that Hakuho obtains the Magaki elder stock first, then inherits Miyagino stock whenever the current stablemaster retires, and (it was already rumoured before) passes Magaki stock to Terunofuji. (Kakuryu I think is expecting to get another one. He is currently on the 5 year temporary membership.)
Otherwise, Hakuho has been very active in promotion of sumo, he ran a yearly children's tournament for example, but the organisation (as you have probably noticed) doesn't like him that much – he's a bit too ambitious... It's possible some will try to set him up a scandal to dishonourably retire him as an elder like what happened to Takatoriki, who was kicked for shamefur dispray of gambling and then borrowing money when he got broke.
I think we first need to see who can make it to ozeki first...
I'd love to see Endo finally getting his shit together, he came from university sumo as a double champion and was hyped by the media, but his performances looked like he's not even trying most of the time. But we've seen him do well in May and September, though sitting out most of July due to injury.
Hoshoryu has a pedigree (his uncle being Asashoryu, before that his family was also involved in bökh in Mongolia), he's doing quite well, and is still young, has room for improvement. I certainly see him getting to ozeki in 2 years, perhaps earlier.
Wakatakakage, also likely future ozeki imo.
Then there's Hokuseiho. If he can get strong and doesn't injure himself, he's almost sure to become ozeki. But it's always a risk with very tall guys. He fucks up his back and he'll end up like Ichinojo.
As for the current and ex ozeki:
Asanoyama isn't yokozuna material, imo, but I can see him getting back to ozeki.
Takakeisho, it depends, his pushy pushy game is too strong, but his arms are too short and anyone who can get close to him will win on the belt. Likely the one thing that will prevent him from yokozuna.
Shodai, well. He's strong, but he gets tricked too easily.
Also, Takayasu, he was an ozeki 2 years ago, but I think his time is up.
From those I think Hoshoryu has best shots at next yokozuna. Come on Japan get your shit together, it's Mongol after Mongol.