Given three unit vectors $u_1,u_2,u_3$ in $\mathbb{R}^3$, can we find some body $K \subset \mathbb{R}^3$ (probably convex) such that the following three things hold

(1) $|P_{u_1^\perp}K|=|P_{u_2^\perp}K|=|P_{u_3^\perp}K|=1$

(2) $|P_{span(u_1,u_2)^\perp}K|=|P_{span(u_1,u_3)^\perp}K|=|P_{span(u_2,u_3)^\perp}K|=1$

(3) $|K|=1$