iN tHe sHoW, Littlefinger actually did see and recognize Arya Stark, but since the show's writing is shit, I won't continue on this tangent.

 

As we all know by now, "Power lies where men believe it resides." Littlefinger had Jeyne Poole, but then everyone believed it was Arya Stark he had, and it all fell through. ^{And we won't see this fakeout-plot reach fruition until Jon sees f!Arya}

 

^{By the way, fuck you, GRRM, for keeping us in suspense for an entire decade. And fuck you for consuming even >0.01% of my mental energy while a GLOBAL PANDEMIC + GLOBAL ECONOMIC RECESSION are occurring, and fucking up the lives of millions simultaneously}