Iwaniec's continuous right-inverse conjecture for the Jacobian

For $p\text{ in }[1,\text{infinity})$, consider the Jacobian operator

$$\operatorname{J}\colon \dot{W}^{1,np}(\mathbb{R}^n,\mathbb{R}^n)\to \mathscr{H}^p(\mathbb{R}^n),$$

where $\mathscr{H}^p(\mathbb{R}^n)$ is the real Hardy space and $\dot{W}^{1,np}(\mathbb{R}^n,\mathbb{R}^n)$ is the homogeneous Sobolev space. Iwaniec's conjecture. For each $p\in[1,\infty)$, there is a continuous map

$$E\colon \mathscr{H}^p(\mathbb{R}^n)\to \dot{W}^{1,np}(\mathbb{R}^n,\mathbb{R}^n)$$

such that

$$\operatorname{J}\circ E=\operatorname{Id}.$$

Thus the Jacobian has a continuous right inverse. The conjecture strengthens the outstanding surjectivity problem for the Jacobian; for $p>1$, the Hardy space agrees with $L^p(\mathbb{R}^n),$ but the supplied text gives no resolution of the conjecture.