There seems to be a $\frac23$-Hölder (or $\frac23$-Lipschitz) onto map $[0,1]^2\to[0,1]^3$. It was apparently found in this paper. I don't know if it has been done elsewhere.
The space filling curves due to Hilbert, Peano and Sierpinski preserve measure and the cited article contains generalizations of these, so it seems quite possible that measure is preserved by some of the corresponding surfaces as well.
I have only been able to access the abstract. It would be great if someone with full text access could check the exact results in the paper.
Here is Fig.12b of the cited paper, showing their Hilbert surface, whose fractal dimension approaches $3$. (*Image added by J.O'Rourke*):
![Fig12Hilbert][1]
