Auxiliary valuation conjecture for partial Stirling functions

Let $5\le\Delta\le8$ and $e\ge5$. For $j\ge0$, define

$$Q_{\Delta,j}=\frac1{(2^e+\Delta)!}\sum_i\binom{2^e+\Delta}{2i+1}(2i)^{2^{e-1}+j}-\frac1{\Delta!}\sum_i\binom{\Delta}{2i+1}(2i)^j.$$

For $j>0$, $j\ne2^{e-1}$, and any integer $m$, define

$$v_{\Delta,j,m}:=e-1-\nu(j)+\nu\left(\frac1{(2^e+\Delta)!}\sum_i\binom{2^e+\Delta}{2i+1}(2i+1)^m(2i)^j\right).$$

Auxiliary valuation conjecture. The quantities satisfy all the following assertions: (a) $\nu(Q_{\Delta,j})\ge e-2$, with the stated residue conditions modulo $4$ for $Q_{7,j}/2^{e-2}$ and $Q_{8,j}/2^{e-2}$; (b) $v_{\Delta,j,m}>\min(\nu(P_\Delta(m)),e-3)$; and, when $(\Delta,m_2)=(7,0)$ or $(8,1)$, (i) $v_{\Delta,j,m}>\min(\nu(P_\Delta(m)),e-2)$, and (ii) if $\nu(P_\Delta(m))=e-1$, the values of $j\ne2^{e-1}$ for which $v_{\Delta,j,m}=e-1$ occur in groups $\{8a+1,8a+2,8a+3,8a+4\}$ for various integers $a$. These detailed estimates are proposed as an auxiliary route to proving the preceding difference-valuation conjecture.