Essential positivity conjecture for radial Toeplitz operators

Let $A^2$ be the Bergman space on the unit disk $4\mathbb{D}4$, and let $T_f:A^2\to A^2$ be the Toeplitz operator with real-valued radial symbol $f\in L^\infty(\mathbb{D})$. Write $\tilde f$ for the Berezin transform of $f$. An operator is essentially positive when its image in the Calkin algebra is positive.

Essential positivity conjecture. The operator $T_f$ is essentially positive if and only if

$$\liminf_{|z|\to 1}\tilde f(z)\geq 0.$$

The preceding corollary proves this criterion when the boundary limit of $\tilde f$ exists. The conjecture seeks the corresponding characterization without assuming that the limit exists; it is motivated by compactness characterizations for Toeplitz operators with measure symbols.