Kaneko–Ohno conjecture on duality differences of multiple zeta-star values

For positive integers $k,n,s$ with $k\geq n+s$ and $n\geq s\geq 1$, let $X_0^{\star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$, and height $s$. Kaneko–Ohno conjecture. For any positive integers $m,n,s$ with $m,n\geq s$, one has

$$(-1)^mX_0^{\star}(m+n+1,n+1,s)-(-1)^nX_0^{\star}(m+n+1,m+1,s)\in\mathbb{Q}[\zeta(2),\zeta(3),\zeta(5),\ldots].$$

The conjecture identifies a duality-type difference that is expected to be expressible using rational polynomials in zeta values, despite general sums $X_0^{\star}(k,n,s)$ not generally having such an expression. Its resolution status is not specified in the source material.