After Triptolemus being added to Hades' jury, is Minos the most important judge?

Question

Minos was the judge of the third and final vote, when the jury was composed of just Rhadamanthys, Minos and Aeacus . After the fourth judge was added, the majority in the judgement was not guaranteed, so I can only see two possibilities here:

1. The vote of Minos (owner of the final vote) will decide if the result is 2-2.
2. Triptolemus is not allowed to be part of the "old" judges cases, neither could they participate in Triptolemus's ones. Making Triptolemus the only judge over the souls of Initiates of the Mysteries.

After Triptolemus being added to Hades' jury, is Minos the most important judge?

---

Edit

I'm trying to know wether I'm wrong about the meaning of the term "final vote" when a 4th judge appears and creates the possibility of a nonconsensus scenario. From my own point of view, Minos was the most important judge, as it was the one who resolved a nonconsensus scenario. Since this is a totally subjective oppinion, and I also may be wrong, didn't want to title this question "...is Minos STILL the most important judge".

Again, I may also be confused because of the fact that Triptolemus owns his own sphere of action and there's no possibility of such a draw, as Codosaur's answer states.

Answer 1

From Plato's Gorgias, page 524:

οὗτοι οὖν ἐπειδὰν τελευτήσωσι, δικάσουσιν ἐν τῷ λειμῶνι, ἐν τῇ τριόδῳ ἐξ ἧς φέρετον τὼ ὁδώ, ἡ μὲν εἰς μακάρων νήσους, ἡ δ᾽ εἰς Τάρταρον. καὶ τοὺς μὲν ἐκ τῆς Ἀσίας Ῥαδάμανθυς κρινεῖ, τοὺς δὲ ἐκ τῆς Εὐρώπης Αἰακός: Μίνῳ δὲ πρεσβεῖα δώσω ἐπιδιακρίνειν, ἐὰν ἀπορῆτόν τι τὼ ἑτέρω, ἵνα ὡς δικαιοτάτη ἡ κρίσις ᾖ περὶ τῆς πορείας τοῖς ἀνθρώποις

translated as:

These, when their life is ended, shall give judgement in the meadow at the dividing of the road, whence are the two ways leading, one to the Isles of the Blest, and the other to Tartarus. And those who come from Asia shall Rhadamanthus try, and those from Europe, Aeacus; and to Minos I will give the privilege of the final decision, if the other two be in any doubt; that the judgement upon this journey of mankind may be supremely just.

The word ἐπιδιακρίνειν can be translated as:

• decide as umpire
• confirm a sentence
• estimate carefully

So Minos' "final vote" as described in Plato's Gorgias should not be interpreted as "tiebreaker". Rhadamanthus judged the dead coming from Asia, and Aeacus those from Europe. They had no "vote" in each others territory. The text seems to infer that Minos would be consulted if the "regional judges" were in doubt.

The addition of Triptolemus, who presided over the souls of Initiates of the Mysteries, is found in some later versions of this myth. As with Rhadamanthus and Aeacus, he is thus given his own "sphere of judgment". The texts that mention Triptolemus as a fourth judge do not mention any exception as to the privilege position of Minos.