One-zero congruence-class conjecture for partial Stirling functions

Let $5\le\Delta\le8$, let $z_{\Delta,\ell}\in\mathbb Z_2$ be the two zeros of $P_\Delta$ indexed by $\ell\in\{0,1\}$, and let $\varepsilon_{\Delta,\ell}$ be $2$ for $(\Delta,\ell)=(7,0)$ or $(8,1)$ and $0$ otherwise. Let $e\ge3$, $0\le m<2^{e-1}$, and assume $m\not\equiv z_{\Delta,m_2}\pmod {2^{e-1}}$. One-zero congruence-class conjecture. There exists $x_m\in\mathbb Z_2$ such that, for every integer $x$,

$$\nu(P_{2^e+\Delta}(2^{e-1}x+m))=\nu(x-x_m)+\nu(m-z_{\Delta,m_2})-1+\begin{cases}\varepsilon_{\Delta,m_2}&e>3,\frac32\varepsilon_{\Delta,m_2}&e=3.\end{cases}$$

This would imply that every congruence class other than those containing the two zeros of $P_\Delta$ contains exactly one zero of $P_{2^e+\Delta}$; the source notes that the complementary classes remain unresolved.