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Recently, a number of counter examples have surfaced where Linear Parameter-Varying (LPV) control synthesis applied to achieve asymptotic output tracking and disturbance rejection for a nonlinear system, fails to achieve the desired asymptotic tracking and rejection behavior even when the scheduling variations remain in the bounded region considered during design. It has been observed that the controlled system may exhibit an oscillatory motion around the equilibrium point in the presence of a bounded constant input disturbance even if integral action is present. This work aims at investigating how and why the baseline Lyapunov stability notion, currently widely used in the LPV framework, fails to guarantee the desired system behavior. Specifically, it is shown why the quadratic Lyapunov concept is insufficient to always guarantee asymptotic stability under reference tracking and disturbance rejection scenarios, and why an equilibrium independent stability notion is required for LPV stability analysis and synthesis of controllers. The introduced concepts and the apparent pitfalls are demonstrated via a simulation example.