The authors of this interesting paper study the fifth-order Korteweg-de Vries (KdV) equation in the normalized form \(u_t+6uu_x+u_{xxx}+u_{xxxxx}=0\) which is a model for small-amplitude gravity-capillary waves on water of finite depth when the Bond number is close to \(1/3\). This model is used to examine the stability of the two symmetric solitary-wave solution branches that bifurcate at the minimum phase speed. In the vicinity of the bifurcation point, the solitary waves take the form of modulated wave packets with envelopes that can be approximated by the same soliton solution of the nonlinear Schrödinger equation, suggesting that both branches would be stable in the small-amplitude limit. It is shown, however, that the branch of the so-called elevation waves is unstable while the branch of depression waves is stable, consistent with numerical results. The coupling between the carrier oscillations and their envelope is essential to this behavior. That is the dimensionless growth rates of the instability modes found for elevation waves are exponentially small with respect to the solitary-wave steepness. By a standard asymptotic procedure the authors discuss the stability of solitary waves with decaying oscillatory tails in other settings.