The isometric-copy translation conjecture for self-affine sponges

Let $K$ be a self-affine sponge, let

$$be its natural probability measure, and let$$

be an isometry such that

$$\big(K\cap (K)\big)>0.$$

Isometric-copy translation conjecture. There exists a translation $t$ such that

$$K\cap (K)=K\cap(K+t).$$

This statement is proposed as a possible tool for extending the results about self-affine Sierpiński sponges from translated copies to isometric copies; its status is left open in the source.