In this thesis - made with the memoir class - the author uses headseplines next to the header content instead of below:

Is it possible to achieve the same with KOMAscript and scrlayer-scrpage, for both the even pages with the chapter title and the odd pages with the section title?
Unlike the linked example I'd like to get:
even page:
chapter title _________________________________________________
odd page:
__________________________________ section number section title
Page numbers are not relevant, as they are in the footer.
Here a MWE to play with:
\documentclass{scrbook}
\RequirePackage[headsepline,automark]{scrlayer-scrpage}
\renewcommand*{\chaptermarkformat}{\scshape}
\KOMAoption{headsepline}{0.1pt}
\pagestyle{scrheadings}
\usepackage{blindtext}
\begin{document}
\chapter{Hello World!}
\section{Section One}
\blindmathpaper
\blindmathpaper
\blindmathpaper
\end{document}
I tried to abolish the headsepline and modify the \chaptermarkformat and \sectionmarkformat using hrulefill, which seems to be a good way. But I can't get it right.
\documentclass{scrbook}
\RequirePackage[automark]{scrlayer-scrpage}
\renewcommand*{\chaptermarkformat}[1]{\scshape \hspace{2ex} \hrulefill}

(chapter name should be on left side, spacing wrong)
With
\renewcommand*{\sectionmarkformat}[1]{\hrulefill \hspace{2ex} \thesection \hspace{1ex} \rightmark }
the odd page looks all right, but it seems to me that it should be done a little different. E.g. why do I need set the spacing manually here: \thesection \hspace{1ex} \rightmark




\def\verrulefill{\leaders\hrule width 0pt height 3pt depth -2.8pt \hfill}% \newcommand{\headseprule}{{\color[rgb]{0.75,0.75,0.75}\verrulefill}}to modify thickness, color and vertical position of the line. Seems a little easier. Or what else is the purpose of the large second part of your answer? – Robert Seifert Apr 19 '15 at 20:00