Turkey of course has a serious conflict of interest.
In two like situations she supports the majority's interests in one case and the minority's in the other, employing two entirely different sets of arguments. Always demonizing the opponent by use of her ultra powerful deep-state propaganda apparatus. Her use of appalling force in the case of Cyprus is meant to ostensibly protect a 12-15% Turkish Cypriot minority when in fact she violates the territorial integrity of a UN/EU state and the human rights of all the Cypriots since 1974, whereas in Turkey itself her appalling use of force is meant to suppress anything Kurdish, for 87 years unabated.
Turkey does not have the political cojones to attend such a conference because she knows her ultra hypocrisy will be unveiled. At such a conference the world will know beyond doubt that the majority of Cyprus, the Greek Cypriots, is light years ahead in her thinking vis-a-vis large ethnic minorities than the majority in Turkey, the ethnic Turks, the latter only just recently awaking to the reality that the Kurds are not Turks and that they want all their legitimate rights that every human being in the world deserves, i.e. social, economic and political rights unhindered by the state!
Such grand unveiling of Turkey's grand hypocrisy is the sheer beauty of the four-party conference idea!
If you would be willing to fund such a conference to take place in Q1-2012, with or without Turkey, please send me an email to email@example.com