Fixed dimension integer programming (both Lenstra's and Barvinok's) uses LLL which guarantees short vectors but not the shortest possible.

Suppose for a given integer programming problem find $x\in\Bbb Z^n$ on condition $Ax\leq b$ where $A\in\Bbb Z^{m\times n}$ and $b\in\Bbb Z^m$ holds we have that the LLL returns optimal basis in polynomial time then are there conditions under which the complexity of the integer programming problem go down from $O(mLn^n)$ to $O(poly(mL)n^{n^\alpha})$ where $L$ is total word size in bits in the presentation of the program at a fixed $\alpha\in[0,1)$?