The mode of classical and free Brownian motion at large time

Let $\mu$ be a probability measure on $\mathbb{R}$ with $\int e^{\varepsilon x^2}\,d\mu(x)<\infty$ for some $\varepsilon>0$. Hasebe and Ueda proved that $\mu*N(0,t)$ is unimodal for large $t$ and asked where its mode is, and the same question for free Brownian motion $\mu\boxplus S(0,t)$. We show that for large $t$ both densities have a unique global maximum, that both maxima converge to the mean $m$ of $\mu$ at rate $1/t$ with explicit constants depending only on $\varepsilon$ and $\int e^{\varepsilon x^2}\,d\mu$, and that \[ M_c(t)=m-\frac{\mu_3}{2t}+O(t^{-2}),\qquad M_f(t)=m-\frac{3\mu_3}{t}+O(t^{-2}), \] where $\mu_3$ is the third central moment; second-order terms are also computed. In the free case uniqueness of the global maximum needs only a finite variance.