Chi-boundedness of graphs containing no cycles with $k$ chords

We prove that the family of graphs containing no cycle with exactly $k$-chords is $\chi$-bounded, for $k$ large enough or of form $\mathrm{\ell }(\mathrm{\ell }-2)$ with $\mathrm{\ell }\ge 3$ an integer. This verifies (up to a finite number of values $k$) a conjecture of Aboulker and Bousquet (2015).