Fluctuation of a planar Brownian loop capturing a large area

We consider a planar Brownian loop $B$ that is run for a time $T$ and conditioned on the event that its range encloses the unusually high area of $\pi T^2$, with $T \in (0,\infty)$ being large. The conditioned process, denoted by $X$, was proposed by Senya Shlosman as a model for the fluctuation of a phase boundary. We study the deviation of the range of $X$ from a circle of radius $T$. This deviation is measured by the inradius ${\rm R}_{\rm in}(X)$ and outradius ${\rm R}_{\rm out}(X)$, which are the maximal radius of a disk enclosed by the range of $X$, and the minimal radius of a disk that contains this range. We prove that, in a typical realization of the conditioned measure, each of these quantities differs from $T$ by at most $T^{2/3 + \epsilon}$.