The current state-of-the-art for proof-theoretic ordinals is somewhere around $\Pi^1_2$-CA$_0$, a subtheory of second-order arithmetic Z$_2$ which in turn is much weaker than ETCS. So at present this is extremely open.
The current state-of-the-art for proof-theoretic ordinals is somewhere around $\Pi^1_2$-CA$_0$, a subtheory of second-order arithmetic Z$_2$ which in turn is much weaker than ETCS. So at present this is extremely open.