In "Bounded Laws of the Iterated Logarithm for Quadratic Forms in Gaussian Random Variables", they at least prove the upper bound in Theorem 3.1.
$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}\leq 1,$$
for $\phi_{n}:=\sqrt{2}E[(V^{n}_{1})^{2}]\log\log(1/E[(V^{n}_{1})^{2})$. And as mentioned in remark 3, we also have
$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}>0,$$
for some special partitions (fast decay $t_{i}=\frac{i}{2^{3n}}$).
Since you are interested in various divisions, here are some more references