Three-dimensional independence conjecture for low-depth 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 $s_2,\ldots,s_p\geq 1$. Low-depth independence conjecture. Among the numbers $1$, $\zeta(2)$, $\zeta(3)$, and $\zeta(2,3)-\zeta(3,2)$, at least three are linearly independent over $\mathbb{Q}$. The source calls this a weaker but very difficult conjecture; it notes that even the independence of $1$, $\zeta(2)$, and $\zeta(3)$ is open, although Apéry proved the irrationality of $\zeta(3)$.