R+L=J - Is Jon snow the son of Rhaegar Targaryen and Lyanna Stark?

The R+L=J theory is a theory that Jon Snow is not Eddard Stark's bastard son but that he is the son of Rhaegar Targaryen and Lyanna Stark. If Rhaegar and Lyanna did indeed get married and that Lyanna gave birth to Jon, then Jon would be a better claim to the Iron Throne than Daenarys. Okay, so the evidence: The last thing that Lyanna said to Ned at the tower of joy before her death was 'Promise me Ned'. Promise what? I hear you ask. Well, Lyanna could be asking Ned to keep a secret of Jon's true identity to protect him from Robert as he killed Rhaegar's other children: Rhaenys and Aegon. Secondly Eddard Stark was one of the most honourable men in the Seven Kingdoms, it seems very unlikely that Ned would father a bastard behind Catelyn's back. In Ned's own words he said to father Jon 'I dishounored myself, and I dishonored Catelyn in the sights of God's and men'. Furthermore Ned states Jon as his blood and not his son, and Eddard had apparently 'lived his lies for fourteen years' and coincidentally Jon is 14 years old at the start of AGOT. Lastly, at the Tower of Joy Lyanna is being guarded by three members of the Kingsguard. This seems very unusual for the Kingsguard to be guarding Lyanna and not Viserys, the heir apparent after Ageon and Rhaeny's deaths, unless they knew about Jon being the son of Rhaegar.