How low of the cartridge output the Allnic H3000 can handle?

Not many here have extensive experience with the H3000, evidently.
why not ask the US distributor?

 Are you using the low output version of the ZYX cartridge? If so those cartridges have about the same output as a platinum version Koetsu. So there should be no problem. Plus, it seems to me that I have read of others who use the low output ZYX cartridges with the H 3000 and are very satisfied.

The 40x SUT in the H3000 will run cartridges down to 0.1mV, provided they have a fairly low number for internal impedance, or DCR. I bought a Benz Ebony TR, tested at 0.14mV with 1-ohm DCR, specifically to take advantage of the Allnic’s design. It is a terrific match. On paper, the same output would work well with a DCR up to around 10. Greater, and you’d probably want to use a separate 30x SUT into the MM inputs.

As a stopgap, I’m using an old Monster Alpha Genesis for the moment- an ancestor of the ZYX--at .2mv output which is pretty low. No issue on my H3000- I did adjust the loading so that it is a little over 100 ohms, compared to my usual Airtights, which load down closer to 70 ohms. A little less gain at that setting- by 2 db I think, but not a major issue. I’m planning on installing a Koetsu next. Should be no issue.
PS: within the range of settings, it is pretty easy to experiment on the Allnic with the twirl of a couple of knobs - mute your phono output temporarily when you change settings. You can easily select the best loading pretty fast, and keep experimenting as time goes on without much trouble.
PPS: did you roll the power supply rectifier on yours, Calvin? That seems to make a big difference in the performance of this unit and there are plenty of anecdotal reports on the web about various rectifier tubes that work. Most aren’t terribly expensive even for "NOS" compared to the price of vintage small tubes these days.
BTW, the importer in the States (if that's where you are) is the lovely Albert Porter, who is a gem and has lots of hands-on knowledge. 

Thank you all of you who has replied, I really appreciate your amswer.
Now I know that it could be my life time phono stage.
@whart I have changed the rectifier tube to Hytron. It sounds good to me but do yoy have any others to recommend?
Thanks,
Calvin