  Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue
consider a system of polynomial equations arising from a sequence of cluster
mutations determined by $D$. They show that any solution gives rise to shape
parameters and thus determines a boundary-parabolic
$\mathrm{PSL}(2,\mathbb{C})$-representation of the knot group. They conjecture
the existence of a solution corresponding to the geometric representation. In
this paper, we show that a boundary-parabolic representation $\rho$ arises from
a solution if and only if the length of $D$ modulo $2$ equals the obstruction
to lifting $\rho$ to a boundary-parabolic
$\mathrm{SL}(2,\mathbb{C})$-representation (as an element in $\mathbb{Z}_2$).
In particular, the Hikami-Inoue conjecture holds if and only if the length of
$D$ is odd. This can always be achieved by adding a kink to the braid if
necessary. We also explicitly construct the solution corresponding to a
boundary-parabolic representation given in the Wirtinger presentation of the
knot group.
