How to evaluate page header at the start of the page?

Questions
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2

I would have a look at how hydra – Typst Universe is dealing with this (with anchors I believe). It may even be working out of the box for this purpose.

Edit: This is a start, adapted from How do I count elements on a page so the result isn't always zero in the header? - #7 by janekfleper


#let exercise(body, label: none) = [
  #show figure: set align(left)
  #set figure(numbering: "1.")
  #show figure.caption: it => context { 
    strong(it.counter.display()) + [ ]
    it.supplement + [: ]
    linebreak()
    it.body
  }
  #figure(kind: "exercise", supplement: "Ex", [], caption: body)
  #label
]

#let disp-exercises() = {
  let ex = query(figure.where(kind: "exercise"))
  ex.filter(it => it.location().page() == here().page()).first()
}

#set page(
  header: context disp-exercises(),
  footer: context disp-exercises(),
)

#exercise(lorem(2))
#exercise(lorem(2))
#exercise(lorem(2))
#exercise(lorem(2))

#pagebreak()
#exercise(lorem(2))
#exercise(lorem(2))
#exercise(lorem(2))