This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to be what Conway, Sloane, and in particular Gabriele Nebe refer to as "covering radius less than $\sqrt 2,$" see

http://www.math.rwth-aachen.de/~nebe/pl.html

and

http://www.math.rwth-aachen.de/~nebe/papers/CR.pdf

One of Pete's relevant questions is

Must a ring which admits a Euclidean quadratic form be Euclidean?

My own observations, confirmed by Pete with the Magma command CoveringRadius, are that the square of the covering radius is rational, and the denominator can always be taken to be a small power of 2 times the determinant of the lattice. Also the power of 2 seems to depend merely on the dimension.

So, that is the question, is the squared covering radius always rational with denominator a (dimension-dependent) power of 2 times the determinant of the lattice?

(note that it may be necessary to have the fraction not be in lowest terms to see the denominator as requested.)