Three-dimensional independence conjecture for selected multiple zeta values

Define the multiple zeta values by

$$\zeta(s_1,\ldots,s_p)=\sum_{k_1>\cdots>k_p\geq 1}\frac{1}{k_1^{s_1}\cdots k_p^{s_p}},$$

for $s_1\geq 2$ and $s_2,\ldots,s_p\geq 1$. Selected-value independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, $\zeta(2,3)$, $\zeta(3,2)$, $\zeta(3,3)$, and $\zeta(3,2,3)$, at least three are linearly independent over $\mathbb{Q}$. The source presents this as a further very difficult conjecture concerning lower bounds for dimensions of spaces generated by selected multiple zeta values; it gives no resolution.