I made a few more tests :
changed the SIM card for a different operator => instability as soon as I activate 4G (LTE)
changed the phone for another Nexus 5 with a brand new battery (replaced this week) => same result
changed the power supply for a bigger one (5A) => same result
So in my case, the problem is independant of :
the phone,
the Slimport hdmi adapter,
the power supply,
the power cables,
the cell phone operator,
the monitor.
I just have to switch on/off 4G data to see the issue appear/disappear. If I am connected to a wifi network, the problem disappears as data is handled through wifi.