Importance of HQ's

[Montana Grizz]

If a unit is within range of all headquarters: Battalion, Regiment, Brigade, Division, Corps and Army, which is the most significant in terms of that unit being provided with supplies?

Specifically, if an engineer platoon is in a given hex and it's parent Battalion HQ is within 5 hexes and it's Regiment HQ's is within 8 hexes and Brigade within 10 hexes, etc., which headquarters is more apt to supply this unit?

Is a unit more apt to be supplied by a Corps HQ than a non-parent Battalion HQ even though they are both within the same hex supply range?

Thanks gentlemen,

Pat