  The Barashenkov-Bogdan-Zhanlav solitons $u_\pm$ for the forced
NLS/Lugiato-Lefever model on the line are considered. While the instability of
$u_+$ was established in the original paper, \cite{B1}, the analogous question
for $u_-$ was only considered heuristically and numerically. We rigorously
analyze the stability of $u_-$ in the various regime of the parameters. In
particular, we show that $u_-$ is spectrally stable for small pump strength
$h$. Moreover, $u_-$ remains spectrally stable until a pair of neutral
eigenvalues of negative Krein signature hits another pair of eigenvalues, which
has emanated from the edge of the continuous spectrum, \cite{B1, BBK, ABP}.
After the collision, an instability is conjectured and numerically observed in
previous works, \cite{B1}.
