Yes, a similar method can.

iWallis[n_] := With[{k = 2 n - 1},
  BoundaryDiscretizeGraphics[
   Table[Rectangle[{x, y}, {x, y} + 1/(k + 2)!!], 
      {x, (k + 1)/(2 (k + 2)!!), 1 - 1/(k + 2)!!, 1/k!!}, 
      {y, (k + 1)/(2 (k + 2)!!), 1 - 1/(k + 2)!!, 1/k!!}
    ]
  ]
]

init = BoundaryDiscretizeGraphics[Rectangle[]];

wallis2D = Fold[RegionDifference[#1, iWallis[#2]] &, init, Range[3]]

bmr = BoundaryMesh[RegionProduct[wallis2D, Line[{{0.}, {1.}}]]];

RegionIntersection @@ (BoundaryMeshRegion[
     MeshCoordinates[bmr][[All, #]], MeshCells[bmr, 2]] & /@ {{1, 2, 
     3}, {3, 2, 1}, {1, 3, 2}})

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Coincidentally, the Demonstration "Cross Menger (Jerusalem) Fractal" was published today, to give more details about the construction. Instead of punching holes through the shape, each iteration is constructed by shrinking and assembling copies of the previous two iterations. (See "Cross Menger (Jerusalem) Cube Fractal" for the difference between using one or two previous iterations.)

Hear, hear, Eric's description is way better than mine at https://robertdickau.com/jerusalemcube.html and https://robertdickau.com/spongeslices.html#asterisk.