"toeing the line" revisited

[Laurence Horn]

We've established, at least to my satisfaction, that "towing the
line" is an eggcorn.  Can anyone confirm or disconfirm the
observation in the commentary track to "The Gathering Storm" DVD (the
2002 HBO production with Albert Finney as Churchill and Vanessa
Redgrave as Clemmie--quite good, I thought) that "toeing the line"
originates (not from athletic endeavors but) from the red line
painted on the floor of the House of Commons, traditionally two
rapier-widths thick, that separates the party in power from the
minority party, to prevent bloodshed on the floor?  Sounds like it
could be plausible, and also like it could be an etymythology, and I
have no way of knowing which.

Larry