The aim of this short note is to present a proof of Conjecture 1.3 of
\cite{CFFR} about the existence of an $A_\infty$-quasi-isomorphism between the
$A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule $K$, introduced in
\cite{CFFR}, and the Koszul complex $\mathrm K(V)$ of $\mathrm S(V^*)$, viewed
as a $\mathrm S(V^*)$-$\wedge(V)$-bimodule, for $V$ a finite-dimensional
(complex or real) vector space.