As far as the rules are concerned, you've already quoted everything that is relevant here:
If the Armor table shows "Disadvantage" in the Stealth column, the wearer has disadvantage on Dexterity (Stealth) checks.
There is no caveat there about "while moving" or "except when standing still", so yes, the penalty still applies when the wearer is standing still.
You are (as always) free to rule otherwise in your games, or if you're a player, ask the GM to rule otherwise. Of particular note here is the description of Dexterity checks:
A Dexterity check can model any attempt to move nimbly,
quickly, or quietly, or to keep from falling on tricky
So if you're standing still, the check might not be a Dexterity check. The description of the Stealth skill doesn't contradict this, either:
Stealth. Make a Dexterity (Stealth) check when you
attempt to conceal yourself from enemies, slink past
guards, slip away without being noticed, or sneak up on
someone without being seen or heard.
Nothing here suggests that standing still should be modeled as a Stealth check, so it could be worth asking your GM (or ruling, if you are a GM), for some other kind of check to model this. The section (I won't quote it all here) on using skills with different ability checks (PHB pg. 175) may be useful for supporting this.