Universal bound conjecture for Gaussian entropic optimal transport

Let $\boldsymbol{A},\boldsymbol{B}\rhd 0$ be positive-definite covariance operators on $\mathcal{H}=\mathbb{R}^{d}$, and let $\pi_{\varepsilon}$ and $\pi_{0}$ denote the entropic and unregularized optimal transport couplings, respectively. Universal bound conjecture. There exists a dimension-dependent constant $C_{d}>0$ such that

for all $\mathbfboldsymbol{A},\mathbfboldsymbol{B}\rhd 0$. The asymptotic scaling is already guaranteed in the non-degenerate finite-dimensional setting. The identity case $\mathbfboldsymbol{A}=\mathbfboldsymbol{B}$ gives $C_{d}\ge d/2$, while numerical experiments suggest rates near $0.36d$ for random full-rank covariances; it remains open whether the identity case is the worst case or whether the smaller observed rates reflect concentration in random matrix ensembles.