Proposition: Unconstrained auxiliary distances

Finite support and a uniform reduction shape do not imply a uniform speed bound in an independently assigned spatial metric.

Proof. Use the independent chains

Assign $T(k,j)=k$ and $X(k,j)=2^{k}j$. Each edge has unit time increment and spatial displacement $2^k\Vert j{\Vert }_2$, which is unbounded. The operational assumptions do not constrain these auxiliary coordinates.