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Is the "mucube" homeomorphic to the loch ness monster?

I'm wondering if two non-compact 2-dimensional manifolds are homeomorphic. The first is the "loch ness monster" a one ended surface formed from the sum of infinitely many tori:

![Loch ness monster illustration](https://math.codidact.com/uploads/p4pmu0wtm1sz40ff7agu7g2ch760)

The other is a surface which I do not know a name for but is related to the "mucube" (a polytope) and so for want of a proper name I am simply calling it that. One way to construct the mucube is to take the tesselation of 3D space by cubes, select all those cubes with at least 2 even coordinates, and then taking the boundary of that set.

Here's a section:

![A section of the mucube](https://math.codidact.com/uploads/wsqsvq85z4eir57xmi233c8o5b6o)

Both of these surfaces are orientable, one-ended, and have infinite genus, so it seems like they might be homeomorphic. I don't have much experience in this type non-compact topology, so I've exhausted all ways I know to tell two surfaces apart. I also can't construct an explicit homeomorphism between the two.

Are they homeomorphic?

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The images in this post are my own work. Both released under CCBYSA 4.0. For an SVG version of the loch ness monster image see [its page on wikimedia commons](https://commons.wikimedia.org/wiki/File:Loch_Ness_monster_surface.svg)