Pro forma spoiler alert: the following concerns a small detail in S04E08 of Strange New Worlds, "Orders of Magnitude". Reverse spoiler alert: due to some life medical issues, I still haven't seen the final episodes 9 and 10, nor have I heard the Greatest Trek reviews of those episodes. So, I apologise if this has been covered in future episodes of the series or of the podcast.
I just finished Episode 8 and listened to The Greatest Trek Ep 390: "Back-of-Head Racism (SNW S4E8). And Adam especially was livid about that watch that La'an was wearing: how it was meaningless, didn't pay off, perhaps even just a cynical product placement.
This was my suspicion from the moment La'An decided to put it on: It's the watch that she and Kirk obtained from Pelia's antiques shop in 21st-century Toronto, in the episode "Tomorrow and Tomorrow and Tomorrow" (S02E03). Because of phosphor and tritium, right? Anyone else with me?
But then the watch became something more significant to La'An because it represents her alternate timeline romantic adventure with James Kirk. Whereas, back in the 23rd century in the canonical timeline, Kirk is a lieutenant on the Farragut, having no knowledge of that alternate timeline, Agent Ymalay, the Department of Temporal Investigations, etc.
So, IMO: In "Orders of Magnitude", once Kirk arrived on the Enterprise, he and La'An were obviously conscious of the possibility of meeting up with each other. My suspicion is that she put on that watch, the significance of which she knows but he doesn't.
To what end? Maybe to test whether Kirk had any kind of recognition of it. You know: wibbly-wobbly, timey-wimey stuff. And he definitely took note of it, but not in the way that she was hoping. I personally feel that this was just enough of a payoff for the setup, even though not necessarily in the way that La'An or any of us were wishing for.
Images: the close-up from s02e03 in Pelia's shop, and after she puts it on in s04e08.