We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $ξ_j$, perturbed by predictable multiplicative factors $λ_j$ with values in intervals $[\underlineλ_j,\overlineλ_j]$. It is assumed that the sequences $\underlineλ_j$, $\overlineλ_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst" sequence $λ_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expectations, and utilizes Hölder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.