Baernstein's extremal exit-time conjecture for schlicht functions

Let ${\cal S}$ be the class of normalized univalent functions on the unit disk, let ${\cal K}$ denote the Koebe functions, and let $K$ be the Koebe function. For $f\in {\cal S}$, write $\tau(f(r\mathbb{D}))$ for the exit time of planar Brownian motion started at $0$ from $f(r\mathbb{D})$, and let $E_0$ denote expectation for this starting point. Baernstein's exit-time conjecture. If $f\in {\cal S}$, $f\notin {\cal K}$, and $r\in(0,1)$, then

$$E_{0}[\tau(f(r\mathbb{D}))^p] < E_{0}[\tau(K(r\mathbb{D}))^p]$$

for any $p\in(0,\infty)$. The same holds when $r=1$ and $p\in(0,1/4)$. This is suggested by Baernstein's strict Hardy-norm comparison and the connection between Hardy norms of analytic functions and moments of Brownian exit times; the stated exit-time comparison is presented as a conjectural consequence rather than as an established theorem.