Let $X, Y$ be connected manifolds and $g$ a Riemannian metric on $X\times Y$ inducing a metric $d$. Is it true that for $x_1,x_2\in X, y_1,y_2\in Y$ one has:
$$\inf_{x\in X}d[(x,y_1),(x,y_2)]≤d[(x_1,y_1), (x_2,y_2)].$$
Some additional (not very useful) remarks:
The conditions of $X,Y$ being manifolds and $d$ coming from a Riemannian metric are likely overkill. If the statement is true it is likely already true for much less stringent conditions, I would guess something like path connected + locally path connected + locally compact is enough.
For product metrics of the form $d[(x_1,y_2), (x_2,y_2)] = d_X(x_1,x_2)+ d_Y(y_1,y_2)$ or $d=\max(d_X, d_Y)$ the statement is trivial.
For a counterexample pull back the standard Riemannian metric under the diffeomorphism $\phi: \mathbb R^2\to\mathbb R^2, (x_1,x_2)\mapsto (x_1,4 x_1^2+x_2)$. The induced metric $d$ is then given by $d(x,y)=d_2(\phi(x),\phi(y))$ where $d_2$ is the euclidean metric. Now consider the points $(0,0)$, $(\frac 1 2,-1)$.