A somewhat sure note on an un-Schur problem

Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur’s theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{1,\dots ,n\}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of $\{1,\dots ,n\}$ lies between $9\mathrm{/}22$ and $8\mathrm{/}15$ asymptotically. Moreover, we study the problem in the general $k$-color setting and establish new non-trivial bounds.