"Does the V20 drive the HE400i"
First, read amir's review of the LG G7. This is the same Quad ess DAC as in the V20. You will see in amir's initial G7 review at the end he commented on output problems. Unknown at the time, the G7/V20 has 3 different 3.5mm output modes, triggered by whatever input device impedance they are feeding. The first, and lowest output, is when an approx 32-50ohm load is detected. Fine for IEMs, not so hot for HE400i. The second next higher 1v output is a 'line out' - set when the +1k or higher input impedance of a typical amp (like the NX3s) is detected. The 3rd and highest 2v output is set when the mis-nomered 'high impedance' is detected - HP of 200 to 600 ohms.
Read the comments on the G7 and other V20 threads and you will see how to set and retain (temporarily until you turn off the V20) in the 2nd and 3rd modes. You need a short 3.5mm male to female extension, and a 200ohm or so 3.5mm 'impedance adapter'.
Bottom line, yes - the V20 will drive the HE400i not badly in 2v 'highnimpedance' mode. But I do not do so. Because it was not designed to do so. While I use SHP9500 directly, I use an NX3s with the HE400i.