Generalized barycentric projection as a Monge mapping for Gromov–Wasserstein distance

Let $cmathbb{S}$ and $cmathbb{X}$ be metric-measure spaces, let $cgamma^*cin\fu0005cGamma^*(cmathbb{S},cmathbb{X})$, and let $cmathcal{T}_{cgamma^*}$ be defined by

$$cmathcal{T}_{cgamma^*}(s):=coperatorname*{\arg\,\min}_{xcin X}cint_X d_X^2(x,x')dcgamma^*_s(x).$$

Generalized barycentric projection conjecture. The map $cmathcal{T}_{cgamma^*}$ is a Monge mapping for the $coperatorname{GW}(cmathbb{S},ctilde{cmathbb{X}}_{cgamma^*})$ problem. The supplied text does not state whether this conjecture has been proved or disproved.