Stretched Brownian Motion: convergence of dual optimising sequences

We consider an irreducible pair $μ\leq_c ν$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglböck, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $μ$ to $ν$ is a Bass martingale, that there exists a dual optimiser $ψ_{lim}$, and the following somewhat surprising convergence result: by adding affine functions, one can make any dual optimising sequence $(ψ_n)_n$ (satisfying some minor technical conditions) converge pointwise to $ψ_{lim}$, save possibly on the relative boundary of the convex hull of the support of $ν$. In the present paper we deal with the more delicate issue of convergence on said boundary, showing in particular that $ψ_{lim}$ is $ν$ a.s. finite, and $(ψ_n)_n$ converges to $ψ_{lim}$ in $ν$-measure.