Correct vertical spacing correction for AMS multline within tcolorbox?

Question

My objective is to put a tcolorbox around one or more AMS multline environment(s). However, there is extra space above the first equation. Presumably I'm running into an issue like this.

I have not found a solution that addresses this situation directly (most seem to address align in tcolorboxes such as this). One solution addresses it indirectly here and if I use -6pt I can get somewhat even vertical alignment. I couldn't find a good way to set this via variable (e.g., \baselineskip).

I have also experimented with empheq but could not get it to behave in the desired way.

My questions:

1. Is there a better way to handle a tcolorbox around an AMS multline directly?
2. If not, is there a better way to do what I've done either by either setting up a custom tcolorbox to always include the "displayskips" and/or not having to hard code the abovedisplayskip value?

A MWE is shown below; my real equation is quite complicated and requires a multline (I'd much rather not try to use an alternative environment because of the amount of equation number management I would need to do in my write up).

Edit to clarify:

I want to emphasize the equations I put inside the box relative to the progression leading up to them and those that follow. So, I want the tcolorbox to respect the typical intra- and inter-paragraph line spacing and not act as an equation but then within the tcolorbox to have even vertical alignment.

---

MWE:

\documentclass{article}

\usepackage{amsmath}
\usepackage[most]{tcolorbox}

\begin{document}

An example \texttt{multline} formula is:

\begin{tcolorbox}
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}
\end{tcolorbox}

\end{document}

---

MWE with workaround:

...
\begin{tcolorbox}
  \abovedisplayskip=-6pt
  \belowdisplayskip=0pt
  \begin{multline}
...

Answer 1

I was testing the \useshortskip command from nccmath, and noticed that simply loading the package seems to solve the problem:

\documentclass{article}

\usepackage{amsmath, nccmath}
\usepackage[most]{tcolorbox}

\begin{document}

An example \texttt{multline} formula is:

\begin{tcolorbox}
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}
\end{tcolorbox}

\end{document} 

Answer 2

You could use the key ams nodisplayskip:

\documentclass{article}

\usepackage{amsmath}
\usepackage[most]{tcolorbox}

\begin{document}

An example \texttt{multline} formula is:
\begin{tcolorbox}[ams nodisplayskip]
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}
\end{tcolorbox}

An example \texttt{multline} formula is:
\begin{tcolorbox}
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}
\end{tcolorbox}

\end{document}

Answer 3

In order to empirically determine the needed correction (expecting it to remain applicable of course for other multline things), I used the device of a millimeter grid. Roughly, one needs -6mm to recover the same vertical spacing as in absence of tcolorbox. I understand from OP this is still too much, then one only needs to increase the 6mm.

\documentclass[a4paper]{article}

\usepackage{amsmath}
\usepackage[most]{tcolorbox}

\def\mygraphpaper{% copied/adapted from https://tex.stackexchange.com/a/311407/4686
  \begin{tikzpicture}
    \draw[line width=.4pt,draw=black!20] (0,0) grid[step=1mm] (\paperwidth,\paperheight);
    \draw[line width=.4pt,draw=red!30] (0,0) grid[step=10mm] (\paperwidth,\paperheight);
  \end{tikzpicture}%
}
\usepackage{tikz}

\usepackage{eso-pic}% has a grid option but not easily customizable

\AddToShipoutPictureFG{\AtPageLowerLeft{\mygraphpaper}}
% advantage to https://tex.stackexchange.com/a/311407/4686 is that
% a single compilation is needed.

\usepackage{geometry}

\begin{document}

An example \texttt{multline} formula is:
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}

An example \texttt{multline} formula is:
\begin{tcolorbox}
\vspace{-6mm}
  \begin{multline}
    f(x) = 1 + 2 + 3 + 4\\
    + 5 + 6 + 7 + 8 + 9\\
    + 10.
  \end{multline}
\end{tcolorbox}

\end{document}

In both cases the distance between baseline above and bottom of x in f(x) appears to be a bit less than 9mm. One could also examine log output of \showoutput... to be more precise.