When Tim had approached him and suggested seeing this world's Superman, Conner had been all for it. He wanted to see with his own two eyes that this world's Clark was here and was okay. Or at least as okay as he could be with what he had been told had happened to him. So he had gone with his two best Bat friends to the museum. He followed their lead in sticking to the shadows, using the range of his vision to help alert them when people were approaching them so that they could hide (not that the pair seemed to need it given the training they had but every advantage was a good thing, right?).
When Tim finally revealed the stasis chamber, Conner could help but remember his time in the Cadmus pod. His hand slightly twitched from the memory but he reminded himself that this was not Cadmus. This was the Bats keeping their friend and ally alive and helping him heal as best they could.
Conner approached the chamber and stared up at the older Kryptonian. He reached out and rested a hand against the surface, his blue eyes full of concern. He had heard Tim saying they had about an hour and he figured that meant the Bat would know they were here without permission. Not that Conner cared. It wouldn't have been the first time he had disobeyed a Batman's orders.
"I'll help keep this world safe Kal," he whispered quietly. "Especially since you're stuck in there." He knew that there was another version of Clark around, and his little brother (that was still something to get use to but Conner liked the idea of family), but he wanted to let this world's Clark know that there was a Super here to help defend it.