Quasisymmetric embeddings of slit Sierpiński carpets

We study the problem of quasisymmetrically embedding spaces homeomorphic to Sierpi\'nski carpet into plane. In case so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms a Transboundary Loewner Property (TLP) which transboundary analogue property Heinonen and Koskela. show that can be embedded plane if only it TLP. Moreover, every $X$ associated "pillowcase sphere" $\widehat{X}$ metric space sphere $\mathbb{S}^2$. embeds quasisymmetric $\mathbb{S}^2$ Ahlfors $2$-regular.