[Thorondir]

They are valid, but illicit because they have declared themselves to not be in union with the Vicar of Christ.

[Land of the Irish]

"They are valid, but illicit because they have declared themselves to not be in union with the Vicar of Christ."

The SSPX has declared no such nonsense:

From their homepage:

"The Society of Saint Pius X professes filial devotion and loyalty to Pope John Paul II, the Successor of Saint Peter and the Vicar of Christ. The priests of the Society pray for His Holiness and the local Ordinary at every Mass they celebrate."

"Oremus pro Pontifice nostro Joanne Paulo.
Dominus conservet eum, et vivificet eum,
et beatum faciat eum in terra,
et non tradat eum in animam inimicorum ejus."