The Barashenkov-Bogdan-Zhanlav solitons u± for the forced NLS/Lugiato-Lefever model on the line are considered. While the instability of u+ was established in the original paper, , 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, [1,2,3]. After the collision, an instability is conjectured and numerically observed in previous works, .