Conjectural generating series for kernels of the matrices $C_k^{(j)}$

Let $k$ range over positive even integers. For $j\in\{1,2,3\}$, let $C_k^{(j)}$ be the square matrices introduced in the paper, and let $\ker C_k^{(j)}$ denote their left-annihilator spaces. Define

$$\mathbb O(x)=\frac{x^3}{1-x^2},\qquad \mathbb E(x)=\frac{x^2}{1-x^2},\qquad \mathbb S(x)=\frac{x^{12}}{(1-x^4)(1-x^6)}.$$

Kernel generating-series conjecture. The following identities are expected:

$$\sum_{k>0:\,\mathrm{even}} \dim_{\mathbb Q}\ker C_k^{(1)}x^k=\frac{1}{x^2}\mathbb S(x)\mathbb E(x),$$ $$\sum_{k>0:\,\mathrm{even}} \dim_{\mathbb Q}\ker C_k^{(2)}x^k=\mathbb S(x)\mathbb E(x),$$ $$\sum_{k>0:\,\mathrm{even}} \dim_{\mathbb Q}\ker C_k^{(3)}x^k=\frac{1}{x^2}\mathbb S(x)\mathbb E(x)+\left(x+\frac{1}{x}\right)\mathbb S(x)\mathbb O(x).$$

These formulas connect left annihilators of the matrices with period-polynomial data and are intended to give upper bounds for spaces of totally odd motivic triple zeta values. Their resolution is not specified in the source.