  We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger
(2022) suggesting that optimal transport with quadratic regularisation can be
used to construct a graph whose discrete Laplace operator converges to the
Laplace--Beltrami operator. We derive first order optimal potentials for the
problem under consideration and find that the resulting solutions exhibit a
surprising resemblance to the well-known Barenblatt--Prattle solution of the
porous medium equation. Then, relying on these first order optimal potentials,
we derive the pointwise $L^2$-limit of such discrete operators built from an
i.i.d. random sample on a smooth compact manifold. Simulation results
complementing the limiting distribution results are also presented.
