EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature.

The answer is likely negative. By [1, Theorem 1.1], the partial sums of any multiplicative function with values $\pm 1$ supported on squarefree numbers are unbounded. As indicated by Will Sawin in the comments, it may be possible to adapt the proof to non-pretentious multiplicative functions with $|f(p)|=1$ ($p$ prime) supported on squarefrees, which applies to $\sqrt\mu$ in particular. However I would be cautious about this without carefully checking the details. The corresponding result for completely multiplicative functions (not supported on squarefrees) is known by Theorem 1.4 of [ \[https://arxiv.org/pdf/1911.06265\]][1].

[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).


  [1]: https://arxiv.org/pdf/1911.06265
  [2]: https://www.ams.org/journals/tran/earlyview/#tran8427