Conjectural period-polynomial relations for double T- and double tilde-T-values

For $N=2$ or $4$, even integers $k\ge4$, and integers $j$ with $1\le j\le (k-2)/2$, define coefficients $a_i$ and $b_i$ by

$$P_{N,k,j}^{(+)}(X+1)=\sum_{i=0}^{k-2}a_i\binom{k-2}{i}X^i,$$

and

$$P_{N,k,j}^{(-)}(X+1)=\sum_{i=0}^{k-3}b_i\binom{k-2}{i}X^i.$$

Here $P_{N,k,j}^{(+)}$ and $P_{N,k,j}^{(-)}$ are the even and odd parts of the polynomial $P_{N,k,j}$ defined from $\widetilde S_{N,k,j}$, and $T(r,s)$ and $\widetilde{T}(r,s)$ denote the corresponding double $T$-values and double $\widetilde{T}$-values. Conjectural period-polynomial relations. The coefficients satisfy

$$\sum_{i=0}^{k-2}a_i\,\widetilde{T}(i+1,k-i-1)=0$$

and

$$\sum_{i=0}^{k-3}b_i\,T(i+1,k-i-1)=0.$$

Moreover, the $\mathbb{Q}$-vector space $V_{4,k}$ spanned by $P_{4,k,j}^{(+)}(X)$ has dimension $\left[(k-2)/4\right]$, the polynomials $P_{2,k,j}^{(+)}(X)$ lie in $V_{4,k}$ and span a subspace of dimension $\left[(k-2)/6\right]$, and the $\mathbb{Q}$-vector spaces spanned by $P_{4,k,j}^{(-)}(X)$ and $P_{2,k,j}^{(-)}(X)$ coincide. The conjectural dimension of this common space $W_k$ is $\left[k/4\right]-1$, and the conjectural number of independent relations among double $T$-values is $k/2-2$.