Computability of the minimal containing-ball radius

Determine whether the parameter $\rho(H)$, defined as the infimum radius whose sufficiently high-dimensional Euclidean balls contain a unit-copy of $H$, is computable from the finite graph $H$.

Background

The parameter $\rho(H)$ controls an exponential upper bound for $#2{^n}{H}$; the paper computes it for odd cycles but leaves its general computability unresolved.