I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$): $$\sum_{k=-\infty}^{-1}\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$

Remark: (1) An easy upper bound is $C\log\lambda$, since $$\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\lesssim \min\{1,\lambda 2^{3k}\},$$ by $|\sin(\lambda t^3)|\le \min\{1,\lambda t^3\}.$

(2) I can verify this uniform bound by Matlab, but I am not able to find a proof.