The fast-embedding quadrant traversal-time asymptotic

Let $(A(t),B(t))$ be the fast continuous-time embedding of the simple harmonic urn, let

$$\tau_f:=\inf\{t\in\mathbb{R}_+:A(t)=0\},$$

and write $\mathbb{E}_n$ for expectation conditional on $A(0)=n$ and $B(0)=0$. Let $\alpha_1\approx -2.0888$ be the constant appearing in Theorem above. Fast-embedding traversal-time conjecture. As $n\to\infty$,

$$\mathbb{E}_n[\tau_f]=\pi/2+O\left(\frac{\mathrm{e}^{\alpha_1 n}}{\sqrt{n}}\right).$$

This asymptotic was motivated by numerical calculations and predicts that the expected time to traverse a quadrant approaches $\pi/2$ with exponentially small error. The supplied text gives no resolution, so the conjecture remains open.