Thanks for the reference made it a little easier
The "inflatable base" would be purely PvE orientated in all modes. It has nothing to do with a station. You do not own a conventional trading market. It doesn't act like a hub of influence. The purpose is a simple act of possession and form of money-sink to throw at a "tent" planted on a moon, in which you store ship/modules/resources (upcoming boosts).
It wouldn't be about owning a corporation, but a place to call "home". This "home" to the owner offers the above mentioned traits. To any other, in any other mode, it appears as an outpost planted on a planet to be raided in your SRV/ship with the objective of obtaining RNG loot ( 60 hunks of palladium because it happens to have a palladium extraction site nearby? ).
This PvE trophy / "home", allows for expansion. Multiple docking bays, larger storage bays, fuel compartments ( thinking of the fuel rats ), a basic defense system.
These would be dynamic and appear as such in every mode. Effectively offering diversity in the sites available to be pirated.
Nothing however contests your entitlement to that base.
A dedicated solo player could in theory make it his objective to invest into such base and start elaborating his "home"plate. Group/open play would allow a wing/group to contribute to this expansion.
Player A group: Hey lets build a tent on moon XX
Player B solo: Hey a tent, lets raid it for some loot!
If both were to meet in open, they would be faced off the same way you and I would be should we encounter in open space. Would I decide to keep you from raiding my base so I can have direct access to my docking bays/storage or do I consider the risk too great and wait for you to leave? After which the base's systems you brought down during your assault would come back online without any further input. You leave with the RNG loot completely independent of my belongings ( hidden storage? ) and I gain full access to lets say switch out my ASP for the Sidey I had stored and take those Crop-harvesters I left to an actual trading market.
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None of this offers dedicated content to a mode.