This is clearly and easily proven untrue.
To preface this, the SHQ has been limited to 30% for SHQ -> Zone and 40% for SHQ -> Unit, though I do not think it matters here because of the limited nature of the logistical branch. Difficulty setting is Hard, if that is relevant.
The 2nd Mechanized Heavy Storm Grenadier Infantry Brigade sits at the very edge of my empire in the newly conquered city of Silverthaw. A road and rail is present, but the rail station has only just finished construction so only a fraction of the massive rail system is available to the city.
A total of nearly 900 logistics points were allocated to the supply of this city and its nearby troops, which has just completed the last 0,23 turns of construction of a new Barracks, Hospital, and HS Rail Station. According to the Items tab 151 Metal and 3 Machines were brought in to assist with construction along with 27 Food brought in for the Workers.
According to the manual all of this should cost 180 weight, although the city seems to take up 361 logistics points. SHQ -> Zone took what it needed and left behind more than enough for the food, and the result?
If Fuel and Energy were free, they would be topped up, but they were not. If Food was the top priority, it would have been topped up first regardless of Fuel and Energy transport costs, but it was not.
Some units have moved this turn, but as logistics take another turn to come online after construction let us see the result before and after ending turn.
The city has taken a lion's share of the logistics, but not all of it; only 653 - the amount available on the city hex itself - of the over 800 points used for the sector were delivered to the Zone, which is why I said I do not think the percentages set at SHQ matter here. There are however enough leftovers to feed the troops, but instead:
Again and despite sitting on the city hex the unit has not topped off Fuel and Energy and is getting a pittance in Food. As you can see stocks are not the issue.
Two things of interest here: one, the only unit that is properly fed is the HQ, although even it was not topped off on Energy and Fuel.
And two, the 3rd Independent Recon Regiment has used 62 logistics points (visible on the After picture), receiving 60 Energy and 2 Food, again suggesting Energy is not free, and also that it is not the direct link to SHQ but something unique to the OHQ that makes it prioritise food deliveries.
Finally, as a bonus to demonstrate costs with Fuel and Energy, let's end turn once more after shuffling the troops a bit to avoid getting cut off by the Farmer swarm.
Over 3000 logistics points used! 1166 for the Recon Regiment alone which sucked up 1132 Energy and 34 Food.
I believe this covers and sufficiently proves my argument.