Inverse-image conjecture for the drop 1 operator and the Hoffman subring

Let $\mathcal{H}=\mathbb{Z}\langle x,y\rangle$, let $\mathcal{H}^0=\mathbb{Z}+y\mathcal{H}x$, let $\mathcal{H}^{2,3}=\mathbb{Z}\langle z_2,z_3\rangle$ with $z_k=yx^{k-1}$, and let $\tau$ be the anti-automorphism defined by $\tau(x)=y$ and $\tau(y)=x$. Let $\mathcal{D}\colon\mathcal{H}^0\to\mathcal{H}^{\geq 2}$ be the drop 1 operator.

Inverse-image conjecture. For any word $w$ in $\mathcal{H}^0$ that cannot be written in the form $w_1\tau(w_2)$ for words $w_1,w_2$ in $\mathcal{H}^{2,3}$, we have

$$\mathcal{D}(w)\notin\mathcal{H}^{2,3}.$$

This conjecture characterizes, in one direction, which words can have drop 1 images in the subring generated by $z_2$ and $z_3$. The paper presents it in relation to the main result and does not give a proof or resolution.