Borwein–Girgensohn parity conjecture for multiple zeta values

Let $r\textgreater 1$ and let ${\boldsymbol{\sl{k}}}=(k_1,\ldots,k_r)\in(\mathbb{Z}_{>0})^r$ with $k_r>1$. The Borwein–Girgensohn parity conjecture. The multiple zeta value $\zeta(k_1,\ldots,k_r)$ can be expressed in terms of multiple zeta values of lower depth when its depth $r$ and weight $k_1+\cdots+k_r$ have different parity. This is a parity result for classical multiple zeta values; the conjecture concerns reducing values of mismatched weight and depth parity to lower-depth values.