For some reason, your file renders with edges even when using a Surface representation (I’m using ParaView 5.12.0-RC3):
My guess is that when you apply the Surface with Edges representation that then the Edges are clashing with the white edges.
If, however, you use the ExtractCellsByType filter and choose Hexahedron for the Cell Type option then the Surface representation doesn’t render the edges:
From there you can then apply the Surface with Edges representation to recover “normal-looking” edges:
NOTE: I see your mesh contains linear hexahedra, but if you ever wish to use higher-order elements, ParaView has a bug / design choice that results in “broken” edges.
If you wish to render higher-order (deg 2+) elements with edges, then for your dataset you would apply an additional ExtractCellsByType filter on the original dataset, this time choosing the Line cell type. Then you would draw the first of these filters as a Surface and the second as a Wireframe:
