  This article determines all the solutions in the finite field $GF{2^{4n}}$ of
the equation $x^{2^{3n}+2^{2n}+2^{n}-1}+(x+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$.
Specifically, we explicitly determine the set of $b$'s for which the equation
has $i$ solutions for any positive integer $i$. Such sets, which depend on the
number of solutions $i$, are given explicitly and expressed nicely, employing
the absolute trace function over $GF{2^{n}}$, the norm function over
$GF{2^{4n}}$ relatively to $GF{2^{n}}$ and the set of $2^n+1$st roots of unity
in $GF{2^{4n}}$. The equation considered in this paper comes from an article by
Budaghyan et al. \cite{BCCDK20}. As an immediate consequence of our results, we
prove that the above equation has $2^{2n}$ solutions for one value of $b$,
$2^{2n}-2^n$ solutions for $2^n$ values of $b$ in $GF{2^{4n}}$ and has at most
two solutions for all remaining points $b$, leading to complete proof of the
conjecture raised by Budaghyan et al. We highlight that the recent work of Li
et al., in \cite{Li-et-al-2020} gives the complete differential spectrum of $F$
and also gives an affirmative answer to the conjecture of Budaghyan et al.
However, we emphasize that our approach is interesting and promising by being
different from Li et al. Indeed, on the opposite to their article, our
technique allows determine ultimately the set of $b$'s for which the considered
equation has solutions as well as the solutions of the equation for any $b$ in
$GF{2^{4n}}$.
