Unless you can get an good high detail image auto trace will never give you the desired results. What I would suggest is a redraw/manual trace.
The way I would do it using Inkscape, use the bezeir to create single lines and build the grid pattern. Then go back and use the square tool to draw a square for the white boxes. Adjust the corner roundness using the square tool. Stretch and skew the square to fill the grid as close as possible, then convert the square to a path and use the node editing tool the get the curve of the sides and fine tune the corner placement.
It's not a hard design project but will be a little tedious.