The polynomial bounded relative K-theory conjecture for a closed solvmanifold group

Let $M^3$ be the closed $3$-dimensional solvmanifold described in the source, let $\pi=\pi_1M^3$, and let $\mathcal P$ be the bounding class of polynomial functions. Let $E\pi$ be the homogeneous bar resolution of $\pi$. Polynomial bounded relative K-theory conjecture. The class

$$[C_\star(E\pi)]$$

represents a nonzero element of infinite order in

$${\mathcal P}K^{\mathrm{rel}}_0(\mathbb{Z}[\pi]).$$

This is proposed as a particular case of the preceding bounded Wall obstruction conjecture, motivated by the cited example in which a degree-two cohomology class is not in the image of the polynomially bounded comparison map. The source gives no resolution of the claim.