Although the author of the article thinks that it is harsh to regard him as a "private heretic", he admits that Pope Honorius was listed as a heretic by two ecumenical councils.
Here is what the Catholic Encyclopedia says:
Pope Honorius was subsequently included in the lists of heretics anathematized by the Trullan Synod, and by the seventh and eighth ecumenical councils without special remark; also in the oath taken by every new pope from the eighth century to the eleventh in the following words: "Together with Honorius, who added fuel to their wicked assertions" (Liber diurnus, ii, 9). It is clear that no Catholic has the right to defend Pope Honorius. He was a heretic, not in intention, but in fact; and he is to be considered to have been condemned in the sense in which Origen and Theodore of Mopsuestia, who died in Catholic communion, never having resisted the Church, have been condemned. But he was not condemned as a Monothelite, nor was Sergius. And it would be harsh to regard him as a "private heretic", for he admittedly had excellent intentions.
CATHOLIC ENCYCLOPEDIA: Pope Honorius I
On one side, it says it would be harsh to call him a "private heretic", since he had "excellent intentions". BUT he was included in the lists of heretics by two ecumenical councils without special remark. The article also says he was a heretic, not in intention, but in fact.
Let's compare this to Pope John XXIII
Pope Honorius was not intentional heretic. It has not be proven that Pope John XXIII was an intentional heretic, either.
Pope Honorius was listed as a heretic in two ecumenical councils. Pope John XXIII was never listed as a heretic by any council.
So if a vacant seat is possible (which I do not believe it is), it would have been vacant at the start of the middle ages under Pope Honorius. The argument is stronger that Honorius was a heretic than it is that Pope John XXIII. Even if Pope John XXIII did join the freemasons, it does not mean that he ever taught heresy. He is guilty of disobeying the commands of the Church to not join the freemasons. If he had done that, I would have to say that this was not intentional. Pope Honorius was not an intentional heretic. And Pope John XXIII did not intentionally disobey the Church - based on the fact that the Church will canonize Pope John XXIII this spring.