In its first year of operation, the Belle II experiment at SuperKEKB collected approximately 10~fb$^{-1}$at the $\Upsilon(4S)$ resonance, with about 100~fb$^{-1}$ expected by the end of 2020. This results in a sizeable sample of $\tau$ pairs, enabling detailed studies including searches for Lepton-Flavor-Violating (LFV) decays. One of the first channels where competitive limits are expected is the $\tau\to e \alpha$ process, where alpha is an invisible Goldstone boson. Here, the currently best limit has been obtained by ARGUS with an integrated luminosity of 475~pb$^{-1}$. Belle II is expected to improve on this result with the data recorded. This contribution will discuss selected analysis details and present first preliminary results and the prospects for future larger data sets.