Since when is a DX claimed spec worth anything? Further, there is no standard by which a vendor has to define the operating voltage range. Is the lower limit when the LED turns on, when the drive current is 1/2 of the rated value in regulation, when the driver reaches full regulation (for what LED Vf?), or something else? We don't know.
Did you even read Post #1 that discusses the DX3256 driver:
"I have ordered these DX
sku.3256 boards about 40 days ago at last I have received them. Theese were totally revised versions, no more using Zetex C310, uses new chip with lots of advantages. I make some measurments. I have measured standart configuration. Add a 1 Ohm paralel resistor to R27 sense resistor and make some more measurments.
At standart config board gives a solid 0,91A from 4V to 16V"
The DX3256 that you get is different from the one shown on the web link. So any operating voltage claims made in the web link should be considered null and void. The driver you got doesn't even use the same IC controller chip.
Did you also read Post #11, which stated that the voltage overhead to run in regulation is about 0.5V? If your have a Vf of ~3.5V, then guess what? You are going to need about 4V min to reach full regulation and 1xLi-ion is probably not a good choice for your batt configuration.
If you use a low Vf emitter like an XP-G, you'll have a much better chance of comfortably reaching regulation with 1xLi-ion. Basically, you want a Vf of no more than 3.2V at ~900mA drive.