The conjecture that the harmonic threshold $3/2$ implies global homeomorphic extension

Let ${\mathbb D}$ be the unit disk, let $f$ be a harmonic mapping in ${\mathbb D}$, and let $\|S_f\|$ denote its Schwarzian norm. A mapping $f$ admits a homeomorphic extension to $\overline{{\mathbb C}}$ when it extends homeomorphically from the relevant domain to the Riemann sphere. Harmonic extension conjecture. If

$$\|S_f\|\leq 3/2,$$

then $f$ admits a homeomorphic extension to $\overline{{\mathbb C}}$. This is presented as a stronger statement than the conjecture that $\sigma_H({\mathbb D})=3/2$, reflecting the proposed distinction between the harmonic and holomorphic settings. The source does not report a resolution.