The answer is negative. This easily follows from [1, Theorem 1.1]. While the cited theorem addresses multiplicative functions with values in $\pm 1$, we may replace each $\pm i$ in the sequence $\sqrt\mu$ with $\pm 1$ and still get bounded partial sums if it had bounded partial sums to begin with. 

[1] Marco Aymone, *The Erdős discrepancy problem over the squarefree and cubefree integers*, Mathematika 68 (2022), no. 1, pp. 51–73, doi [10.1112/mtk.12117](https://doi.org/10.1112/mtk.12117).