A short note on the Bruiner–Kohnen sign equidistribution conjecture and Halász’ theorem

In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Hal\'asz' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.
Comment: 4 pages, main result made unconditional, minor changes due to referee's reports