Eisenstein-series identities for the rank-two and rank-three Witten zeta functions

Let $n$ be a positive integer, let $f_B(x)=(1+x)(1+2x)$ and $f_G(x)=(1+x)(1+2x)(1+3x)(2+3x)$, and let $E_{2k}(\tau)$ denote the Eisenstein series for $\tau$ in the upper half-plane, with $E_k(\tau)=0$ for odd $k\geq 3$. For a power series $h(x)$, write $[h(x)][x^j]$ for the coefficient of $x^j$ in $h(x)$. Eisenstein-series identity conjecture. For every positive integer $n$, the following two equalities hold:

$$\frac{\zeta(-8n-1)E_{8n+2}(\tau)}{3}(1+2^{-1-4n})\times [f_B(x)^{2n}\log f_B(x)][x^{1+6n}] =\sum_{i=0}^{4n}[f_B(x)^{2n}][x^i]\zeta(-i-2n)\zeta(i-6n)E_{2n+i+1}(\tau)E_{6n-i+1}(\tau).$$ $$\frac{\zeta(-12n-1)E_{12n+2}(\tau)}{5}(1+3^{-1-6n})\times [f_G(x)^{2n}\log f_G(x)][x^{1+10n}] =\sum_{i=0}^{8n}[f_G(x)^{2n}][x^i]\zeta(-i-2n)\zeta(i-10n)E_{2n+i+1}(\tau)E_{10n-i+1}(\tau).$$

These identities generalize the corresponding vanishing relations for the single-variable Witten zeta functions of types $B$ and $G$; they have been checked computationally for $n\leq 10$, but no proof is provided here.