#### Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $0 \le \sigma \le \frac12$

By joro

Based on limited numerical evidence, I suspect this conjecture.

Conjecture: Fix $$0 \le \sigma \le \frac12$$ and let $$t > 0$$. Between consecutive local extrema of $$\Re \zeta(\sigma+i t)$$ (resp. $$\Im \zeta(\sigma+ it)$$), there is always a zero of $$\Re \zeta(\sigma+i t)$$ (resp. $$\Im \zeta(\sigma+ it)$$).

Verification for several random $$\sigma$$ and $$0 < t < 30000$$ and for a few random intervals didn't show any counterexamples.

For $$\sigma > \frac12$$ it is false and on the other hand this appears counterintuitive to me.

Counterexamples? (please check for closely spaced zeros that might look like a single local minimum on a large plot).

Even if it is true, a conditional proof probably will be hard yet welcome.

For Siegel $$Z$$ function on the critical line RH implies this for $$t$$ large enough.

Maybe can be generalized to $$\sigma \le \frac12$$.

Plot of a random interval: