^^^ if that was the case, why did he do the Rajsuya Yagya and not Ashwamedh?? In fact as per KMG after the Digvijay, even Karna says to Duryodhan that the entire country is under you, although we know that wasn't possible (I know it's an interpolation but still there Seems some wordly differences)
Aside if all the kings were actually under Yudhishtir's suzarainty how could they speak against their decision on Pratham Pujya to nearly destroy it.
We need to check the actual word in Sanskrit that would make it clearer