Conjecture on simple real zeros of multiple zeta-functions

Let $r\geq 2$. For $2\leq k\leq r$, let $I_r(k)$ denote the number of inter-admissible zeros of $\zeta_r(s)$ in the interval $(1/k,1/(k-1))$, where an inter-admissible zero is a real zero in this interval. Conjecture on the real zeros. For any $r\geq 2$, all inter-admissible zeros of $\zeta_r(s)$ are simple, and

$$I_r(k)=\left[\frac{r}{k}\right]\qquad (2\leq k\leq r).$$

The numerical data for $2\leq r\leq 10$ support this conjecture, but no rigorous proof is known.