How to make Figures logically belonging to a long theorem-like halmos-terminated environment float within the environment and keep order

Question

Let's assume that in a huge book of over 500 pages you often typeset long Notes (or other theorem-like environments) with Figures. You want that your Figures possibly stay near the declared places and that if a Figure floats at all, it floats within the Note logically containing it, i.e., between “Note ⟨note number⟩ ⟨(possibly, Note title in parens)⟩” and ■ terminating the Note. You also want that the floating and non-floating figures together maintain the same total order in the output as in the input. We could either nail the last figure of a Note with [H] and put a \FloatBarrier right before it (which makes the figure non-floating) or have the last figure floating but a halmos way too low:

\documentclass{svmono}% V5.10 from https://resource-cms.springernature.com/springer-cms/rest/v1/content/20566/data/monographs .  For testing purposes, I hope, book would do as well.
\usepackage[latin,USenglish]{babel}%%% solely because lipsum is Latin
\usepackage{amssymb}
% \usepackage{float,afterpage}% also see \flushhere from http://tex.stackexchange.com/a/246315
\usepackage{placeins}
\usepackage{lipsum}
\makeatletter
% typesetting halmos:
\newlength\lastSkipOfEndCenter
\newcommand*{\halmosEndingNonProofAfterCenter}{\setlength\lastSkipOfEndCenter{\lastskip}\removelastskip\nopagebreak\par\nopagebreak\vspace{\dimexpr-\baselineskip-\parskip\relax}\nopagebreak{\unskip\nobreak\hfil\penalty50\hskip1em\null\nobreak\hfil\(\blacksquare\)\parfillskip=\z@\finalhyphendemerits=0\endgraf}\vskip\lastSkipOfEndCenter}
\makeatother
\begin{document}
\chapter{Chapter}
\newcommand{\figureheight}{19ex}% With ≤17ex, the halmos appears way too low below the figure. With 18 ex, the halmos appears of the next page.  With ≥19 ex, Fig. 3 goes to the next page and the halmos appears way too low.
\lipsum[1]
\begin{figure}[htbp]\centering
  \rule{10em}{\figureheight}
  \caption{1}
\end{figure}
\lipsum[1]
\FloatBarrier
\begin{note}% Note 1
  \lipsum[1]
\begin{figure}[htbp]\centering
    \rule{10em}{\figureheight}
    \caption{2}
  \end{figure}
\lipsum[1]
%\FloatBarrier
  \begin{figure}[htbp]\centering%%% We don't wish to simply use [H] for this figure and [h] elsewhere in the chapter because then the Figures might hypohetically come out in the wrong order. Otherwise, and with \FloatBarrier right before, [H] seems to do the job. Using [H] everywhere in the chapter is not an option because outside this environment, LaTeX does a good automatic job of placing the environment, and a manual placement everywhere would have to be reviewed too often.
    \rule{10em}{\figureheight}
    \caption{3}
  \end{figure}%
  \FloatBarrier
  \halmosEndingNonProofAfterCenter
\end{note}
\lipsum[1]
\begin{figure}[htbp]\centering
  \rule{10em}{\figureheight}
  \caption{4}
\end{figure}
\lipsum[1]
\end{document}

Either of the two solutions probably preserves the order (though, as Skillmons explains in https://tex.stackexchange.com/a/654613, \FloatBarrier is not a panacea); feel free to correct me if I'm wrong. Is there any way to have the last Figure of the Note (in our example, Figure 3) floating (in our example, only above ■) such that the halmos is correctly placed on the last line of the Note, whether it is text or caption (in our example, of Figure 3)? Is there further any way to avoid the page break right before the halmos (to test, set \figureheight to 18ex)?

Answer 1

Here your figures are logically part of your note and you don't want them to float out so I think the cleanest solution is likely to be to use [H] non-floating figures in the note and use \Floatbarrier in the definition of \begin{note} so figures are always set in order.

The worst "bad" thing that will happen is that a note is forced to start a new page if pending floats are flushed, and as there are no floats within the note you know whether the final visual line comes from normal text or a caption, so can place the halmos qed appropriately.