deducting harmonic suppression specification?

I was wondering, if there is a commercial 2.4GHz antenna how would it perform in harmonic frequencies?

Here's an example 2.4GHz monopole antenna.

I checked the specs but it doesn"t give any info on harmonic frequencies.

Based on what I learned in basic antenna theory, monopoles also radiates well in harmonics right?

Does this theoretical knowledge also apply well on commerical products such as above?

IOW, does these commerical monopole antennas provide some harmonic suppression?

Yes, antennas can be resonant at harmonics - although there's more to making an antenna work than just the "resonance".

Is what you have a "pure" monopole, or is there a tuning/matching network inside? If so, the matching network might crush any harmonics.

Antenna alone does not generate harmonics. Any antenna can radiate harmonics from a signal source connected to it. Antennas usully do not come with specifications indicating the capability of radiating harmonics. Some wideband antennas (contrary to the shown monopole) can radiate harmonics better than resonant antennas.