The conjecture that the harmonic univalence threshold of the unit disk is $3/2$

Let ${\mathbb D}$ be the unit disk, let $\sigma_H({\mathbb D})$ denote the harmonic univalence threshold of ${\mathbb D}$, and write $\|\cdot\|$ for the Schwarzian norm on ${\mathbb D}$. Harmonic threshold conjecture.

$$\sigma_H({\mathbb D})=3/2.$$

The conjecture is motivated by the harmonic mapping $f_\alpha$ and proposes the sharp threshold at which injectivity can fail in the harmonic setting. The source does not report a resolution.