Conjecture on simple inter-argument zeros of multiple zeta-functions

Let $r\geq 2$. For $2\leq k\leq r$, let $I_r(k)$ denote the number of inter-argument zeros (IAZs) of $\zeta_r(s)$ in the interval $(1/k,1/(k-1))$. Simple IAZ conjecture. All IAZs of $\zeta_r(s)$ are simple, and

$$I_r(k)=\left[\frac{r}{k}\right]$$

for $2\leq k\leq r$. Numerical data for $2\leq r\leq 10$ support the conjecture, but the authors state that they have found no rigorous proof.