I could see a Tri-Core and a Core, but I highly doubt Spiral being able to have more than one Trooper with Tri-Core in a list.
The latest information we have is that only 1 Tri-Core can be active at a time. Still no info on if you can also have a regular Core.
I'd assume that only one "Core" will be allowed on the table, of any Tri-Core / Fireteam: Core kind. That's most logical to me from the design perspective. It's effectively a F: Core subtype, one that allows for the 5-man effect with 3 models only.
What is a bigger question is what the switching requirements are for different tri-core links. Does it cost a command token? Is it a short skill to activate? Can you switch any time the triad is activated?
It costs a SWC as well, so you're capped at one unless you are planning on taking really cheap SWC weapons and no minelayer/indirect control/kaeltar.
To complicate it even more. Do you need to state that Triad X is Tri-Core? This gets relevant if you Holoproject the Taagma as something else. "Oh, that Kumotail was a Taagma and now that Kossuil will launch her ARO Panzerfaust as in a 5-man link? Stop wrecking my math will you!" *boom*
The way I understand that passage from the article, Fireteam: Core was not mentioned in any way. While it's possible that this limitation will be added, speculations on the matter of inability to combine a Core with Triad buffed by Tri-Core right now lay somewhere between "I think that would be broken" and "You can't have slaughter without laughter because you cannot spell one without another". Anyway, I'm more curious what kind of bonuses Tri-Core will give to a 2-man Triad.
My assumption is none. As far as I've understood, Tri-core is supposed to provide 5-man bonusses while maintaining the fragility of a three-man-link.