Conjecture on the Hilbert symbol and the coefficients $U_k^{(u)}$

Let $\ell$ be the prime and retain the notation introduced for $J$, the Hilbert-symbol computation, $U_k^{(u)}$, $U_k'$, $H$, $c_N'$, $b$, $\pi$, $\Delta$, and $\zeta_{\ell}$. For each $u\in\\{0,1\\}$, there exists an integer $r_u$ with

$$0\leq r_u\leq \left\lceil\frac{j-i_0}{2}\right\rceil.$$

The conjecture. For $0\leq k\leq \left\lceil\frac{j-i_0}{2}\right\rceil$, the coefficients satisfy the stated congruences for $U_k^{(u)}$ and $U_k'$, and consequently

$$H/\ell^{N-\operatorname{ord}_{\ell}(c_N')}\equiv \begin{dcases} 1 \pmod{\ell} &(1\leq \operatorname{ord}_{\ell}(c_N')<N)\\\\ -1 \pmod{\ell} &(\operatorname{ord}_{\ell}(c_N')=N) \end{dcases}$$

and

$$(1+\pi^{f_{\Delta,\pi}-1},\Delta)_{\ell^N}=\begin{dcases} \zeta_{\ell}^{2b(-1)^{N-1}} &(b\neq0)\\\\ \zeta_{\ell}^{\frac{2c_N'}{\ell^{\operatorname{ord}_{\ell}(c_N')}}} &(b=0,\\ 1\leq\operatorname{ord}_{\ell}(c_N')<N)\\\\ \zeta_{\ell}^{-\frac{2c_N'}{\ell^{\operatorname{ord}_{\ell}(c_N')}}} &(b=0,\\ \operatorname{ord}_{\ell}(c_N')=N). \end{dcases}$$

The author reports verification for $N=2$, $\ell\leq50$, $N=3$, $\ell\leq20$, and $N=4$, $\ell\leq20$; the result is proved in the case $\operatorname{ord}_{\ell}(c_N')=N$, but remains open in general.