Conjectured Wigner 6j-symbol summation identities

For fixed $N$, let

$$\mathcal{W}^{ij\ell}=\left\{\begin{matrix} i & j & \ell \\ \frac{N-1}{2} & \frac{N-1}{2} & \frac{N-1}{2}\end{matrix}\right\},\qquad \mathcal{W}_j^i=\left\{\begin{matrix} i & \frac{N-1}{2} & \frac{N-1}{2} \\ j & \frac{N-1}{2} & \frac{N-1}{2}\end{matrix}\right\},$$

let $\lambda_i=i(i+1)$, and let $H_j=1+1/2+\cdots+1/j$ be the $j^{\text{th}}$ harmonic number. The conjectured Wigner $6j$ identities. For all $1\leq j,\ell\leq N-1$, the four displayed summation identities in the source hold, with the first subject to $j\neq\ell$:

$$\sum_{i=1}^{N-1}\frac{2i+1}{\lambda_i}(\mathcal{W}^{ij\ell})^2=\frac{1}{N|j-\ell|(j+\ell+1)},$$ $$\sum_{i=1}^{N-1}(-1)^i\lambda_i(2i+1)(\mathcal{W}^{ij\ell})^2=(-1)^{N+1}(\lambda_j+\lambda_\ell)\mathcal{W}_j^\ell,$$ $$\sum_{i=1}^{N-1}\lambda_i(2i+1)(\mathcal{W}^{ij\ell})^2=\frac{(N^2-1)(\lambda_j+\lambda_\ell)-2\lambda_j\lambda_\ell}{N(N^2-1)},$$ $$\sum_{i=1}^{N-1}\frac{2i+1}{\lambda_i}\left(\frac1N+(-1)^{i+j+N}\mathcal{W}_j^i\right)=\frac{2H_j}{N}.$$

These identities are proposed from recurring numerical patterns in the Ricci-curvature computations and extend standard Wigner $6j$ summation formulas; the source gives no resolution beyond numerical evidence.