Can the product of two $\mathsf Y$'s be embedded in 3-space?

The product space consists of 9 rectangular sheets, glued together along two of the edges of each sheet. There are 6 glue-edges where three sheets meet, and all of these edges meet at a center point.

Assume we have an embedding and intersect it with a sphere around the center point that is small enough not to contain any points on the boundary of the embedding.

There are 6 points on the sphere that represents glue-edges (just pick an arbitrary one if the glue-edge crosses the sphere multiple times), and the sheets themselves cross the sphere in curves between these points.

Together, the whole figure drawn on the sphere is the complete bipartite graph $K_{3,3}$ -- but that is well known not to be planar, so it can't be drawn on a sphere without intersections!