Not exactly their war effort.
The Triglavians are not one cohesive empire, but rather 3 sub factions that are constantly in conflict with one another, a thing they call “The Struggle”
They also hold the idea of survival of the fittest to a near-religious level. Every idea, person, plan, or note must be validated through a “Proving”.
When they found Sansha they detected Hivelinking, which they are staunchly against for reasons handed down to them from ancient times. Also, Sansha tried and in some cases succeeded in corrupting Triglavian’s people and equipment.
However, when they found us they scanned us and detected no hivelinking. Instead of extirpation, they invoked “Cladistic Proving”. Cladistics, or Clades is a biological term meaning 2 sub-groups of a Biological species. So “Tiger” and “tabby cat” would be Cladistically related. They see us as like them, and as such we are permitted to prove ourselves to them, and thus be found fit to be entered into the “Flow of Vyraj” (Their collective sense of destiny or purpose) and even enter the Domain of Bujan (a place we have not seen yet). And they perform this proving by fighting us. Glorification to the fit, and mortification to the unfit shall unfold from the flow.
However, they have a mission in Pochven and have declared any attempt to disrupt their efforts and “mortify” their flow will be declared Poshlost (corrupt) and extirpated. Zorya Triglavs actual title is Detached Executive Troika for the Sublimation of Poshlost Flow. If you work with them, they will accept you in and you can begin the process of Proving as an Aspirant Noradnya and/or Sobornost Kybernaut.
So that is why you lose standings when you encounter the Triglavians outside of the abyss, but not inside.
The bit about the war effort is that those who DO aid in “guiding and grounding the flow” shall be glorified and permitted into Bujan, whatever that means. And the instructions we have to aid is to patrol Pochven and eleminate any Poshlost intruder (basically anything not loyal to the Triglavians.
Heck right now in Pochven the Triglavians are fighting more with each other than anyone else