Coupling-convexity of a diffusion with respect to its starting position

In the present paper, we consider a $d$-dimensional time-homogeneous stochastic differential equation with diffusion matrix $a$ and no drift coefficient. We check that the convexity in the starting position $x$ of the distribution of its solution on the path space when the codomain is endowed with the convex order is equivalent to the convexity of ${\mathbb{R}}^d\ni x\mapsto \sqrt{{\zeta }^{\ast }a(x)\zeta }$ for each $\zeta \in {\mathbb{R}}^d$. To do so, we introduce the seemingly stronger notion of coupling-convexity in the starting position and check that it is also equivalent to the convexity of the above square-rooted functions.