A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Hal\'asz' 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.