An interesting test is to set the “Maximum Discharge Power” to 0 for the battery. This stops the battery from discharging, as can be seen by the “Battery discharge” line in the FusionSolar graphs dropping to 0.
However, my experience has been that there is a sizable auto-discharge anyway. You can see the graph of the Battery SoC of my 5kWh LUNA battery here between 17h and 23h discharging from 71% to 63%. This means an auto-discharge of 1.33% per hour 
The “Battery day discharge” sensor reports some discharge though: from 0.06kWh to 0.47kWh (as does “Daily Yield”: from 1.90kWh to 2.14kWh, note: a battery discharge of 0.41kWh results in a daily yield increase of only 0.24kWh):
Is this because it requires energy for the inverter to stay ‘on grid’? It that increase in ‘Daily yield’ really injected into my home electricity net?
It really doesn’t bode well for the overall profitability of my home battery 
. Say that I want to use it in winter only for reducing my peak grid usage (which is taxed in Belgium starting from this year), then it will cost me 32% of my battery SoC just to be available and do nothing, that’s a loss of 1.6kWh per day. Even if (0.24/0.41 =) 58% ‘leaks’ back into my home electricity net (which I doubt), this is a very poor performance.