^^ PsiBorg, Agreed.

100%
In Season 4, episode 8… Some of you are forgetting that during his trial in the Vale, Littlefinger told Lord Royce and Lady Waynwood that Lysa Arryn
COMMITTED SUICIDE.
Skip to the 50 second mark:
And that's the story he's been
STICKING TO for several years.
Now in the Season 7 finale…. he ADMITS to pushing her & killing her when he says: "Sansa, I did it to PROTECT YOU."
Skip to the 2:20 mark. Listen to the VERY FIRST question that Sansa asks him.
That’s the Whole Ballgame.

Right there. End of Story.
That ALONE was enough for the Starks to execute him… for MURDERING their Aunt Lysa.
Didn't matter 'what' Littlefinger said AFTER that.
Everything else that the Starks had accused him of doing....
'could be assumed to be true'... if Littlefinger had lied FOR MANY YEARS about the facts concerning Lady Arryn's death.
That's WHY Lord Royce said:
"I think not" 
when Baelish demanded he be taken SAFELY back to the Vale.... because Lord Royce would've ALSO executed him for what he did to Lysa Arryn alone, when he had enough power at the Vale (back then) to carry out the sentence himself.