Most Circular Partition of a Square

What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? The circularity of a polygon is the ratio of the radius of its smallest circumscribing circle to the radius of its largest inscribed circle. Thus circular pieces have circularity near $1$, and noncircular pieces have circularity greater than $1$. An optimal partition minimizes the maximum ratio over all pieces in the partition.