I wrote up a short note, which answers the following question for the two standard simplicial models of the sphere: for which $d$ are they $d$-Segal?
The two models (let's say for $n \geq 1$) are $\Delta^n / \partial \Delta^n$, the $n$-simplex with its boundary collapsed to a point, and $\partial \Delta^{n+1}$, the boundary of the $(n{+}1)$-simplex. The answers are different!
Theorem. For $n\geq 1$, $\Delta^n / \partial \Delta^n$ is $2n$-Segal, but not lower $(2n{-}1)$-Segal. If $n > 1$, then it is upper $(2n{-}1)$-Segal if and only if $n$ is odd.
Theorem. For $n \geq 1$, $\partial \Delta^{n+1}$ is $(n{+}1)$-Segal but neither upper nor lower $n$-Segal.
(In my paper with Justin last year, we showed that $\Delta^n / \partial \Delta^n$ is lower $(4n{-}1)$-Segal, so this is an improvement.)
Just as a warning: I've included no background at all in the note, and it's probably not readable to anybody but me right now. I was mostly thinking about including the first theorem above as an application of the main theorem of a paper that I'm about to post – but it turns out I didn't need my theorem at all! So I've just made a separate doc. Once that preprint appears (edit: here is the arxiv link), you'll be able to read the note properly. Here's the main theorem of the other paper, which generalizes a recent theorem of Walker Stern for $d=2$:
Theorem. Let $X$ be a simplicial set. If $X$ is $(d+2)$-coskeletal and satisfies the upper (resp. lower) $d$-Segal conditions in the first two relevant simplicial dimensions, then it's upper (resp. lower) $d$-Segal. Conversely, every upper or lower $d$-Segal simplicial set is $(d+1)$-coskeletal.
I probably wouldn't have completed the calculation in the note if I hadn't been working on this coskeletality result. But I was sitting in Federal Hill Park during the Sunday between CT2026 and DV60, and decided to ask Claude Fable to take on some calculations for me (I figured I'd try to get some use out of it before I lost access). The coskeletality theorem turns the question into a finite problem, perfect for brute forcing with Python, so I tapped tapped tapped into the iPhone app and let Fable spin while I walked around a bit. This ended up giving me the answers for $\Delta^n/\partial \Delta^n$ when $n$ was not so big. There's a bit more about AI use in the note!