Does anyone know what is the fewest-piece dissection of the surface of a regular tetrahedron to the surface of a cube (of the same area)?

It is well-known that the volume of a regular tetrahedron cannot be dissected to the volume of a cube, because their Dehn invariants differ. But their surfaces have a dissection, by applying the Boyai-Gerwien theorem. Applying that theorem in one way (among several options) leads to an $31$-piece dissection (if I calculated correctly), illustrated below. But this is surely very far from optimal (and hardly aesthetically pleasing). Likely the question has been explored, but I have not found any literature. It could make a pleasing contrast to the impossibility of a volume dissection.

Let the cube edge length be $1$, so its surface area is $6$. A tetrahedron edge length of $L= 2^{\frac{1}{2}} 3^{\frac{1}{4}} \approx 1.86$ leads to a surface area of $\sqrt{3} L^2 = 6$.

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Surface dissection of regular tetrahedron to cube.
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Related: "Covering a Cube with a Square."