What does Gertrudes statement “The lady protests too much, methinks” tell us about her possible guilt in her first husband’s death?

Gertrude's comment "The lady protests too much, methinks" in act 3, scene 2, of Shakespeare's Hamlet exposes her own guilty conscience.
"The lady" that Gertrude is referring to is the player queen. Gertrude makes this comment while watching, along with the rest of the castle, the play that Hamlet has devised for the players to perform, which mirrors the events of his father's murder in hopes that Claudius's response to the uncannily familiar performance will finally confirm Hamlet's suspicions and incriminate his uncle. The player queen is Gertrude's analog in the plot, a caricature of herself, so when Gertrude comments on the other woman's actions her comment reflects back on herself.
The sentiment of "The lady protests too much" is essentially that someone can deny something so many times that it becomes unbelievable. Throughout Hamlet, Gertrude claims ignorance of her new husband/past brother-in-law's vile actions. But here, the audience gets the sense that maybe Gertrude has been protesting too much herself. Maybe she does have her suspicions about what happened to Hamlet's father, and the reason she protests so much is to keep her own doubts at bay.