But does Armenia accept the details of the proposals as outlined by President Aliev?
Mr Aliev's triumphal return from America to Azerbaijan this summer was an extravagant affair.
One Baku theatre director has no fewer than eight portraits of Mr Aliev in his office.
Dr Aliev suggested the material might one day be scaled up to hide large objects including military vehicles.
The real test is for Mr Aliev, who is as much of a Caucasian enigma as his country.
ECONOMIST: Azerbaijan's election: A Caspian contradiction | The
Mr Erdogan was expected to try to square Azerbaijan's president, Ilham Aliev, in a visit to Baku this week.
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This is Mr Aliev's chance to prove that he can become a truly democratic leader, not just a simulacrum.
ECONOMIST: Azerbaijan's election: A Caspian contradiction | The
Presidents Aliev and Shevardnadze are doing their best to pre-empt these dangers.
For now Mr Aliev's regime, which is already working on a bid for the 2020 Olympics, can assume that its opponents offer no threat.
No sooner had she sat down with Mr Aliev in his seaside palace, lit by giant chandeliers, than he vented his anger at Armenia.
When Mr Aliev returned to power in 1993, after Azerbaijan's first two post-Soviet leaders had failed to entrench themselves, he dropped all proletarian pretension.
Azerbaijan's president, Ilham Aliev, has been sabre-rattling to drum up support.
But the other Mr Aliev is secretly in cahoots with them.
ECONOMIST: Azerbaijan's election: A Caspian contradiction | The
President Aliev, whose regime is growing harsher in the face of rising dissent, would love to make a triumphant announcement that the Ceyhan route was finally assured.
According to Mr Aliev, his Armenian counterpart Levon Ter-Petrosian told him when they met in Strasbourg earlier this month that Armenia also, basically, accepts the Minsk Group proposals.
Sensing the people's grim mood, Mr Aliev rarely travels round his own country, preferring to make grandiose trips abroad: this month, he is due to make a state visit to Britain.
At a recent World Economic Forum meeting in Warsaw, where the three Caucasian presidents met, Mr Aliev dismissed any talk of freeing up trade until Armenia stopped occupying part of his country.
Last month, a former deputy interior minister under the country's first post-Soviet president, Albufaz Elchibey, whom Mr Aliev ousted five years ago, was arrested after a grenade was allegedly found in his car.
The oil, competition with Russia for influence in the former Soviet Union, plus Azerbaijan's strategic location, might suggest that America and the European Union should be content with Mr Aliev and the stability he seems to offer.
Last week the Azerbaijani president, Heidar Aliev, said this second stage would require the Armenians also to give up the Lachin corridor and the town of Shusha, within Karabakh, which before the war had a predominantly Azeri population.
Since Heidar Aliev, now 76, had a heart operation five months ago in the United States, two museums devoted to his genius have opened, one in the Azeri exclave of Nakhichevan, where he was brought up, the other in a suburb of Baku, Azerbaijan's capital.
This Mr Aliev seems to understand that, since power in Azerbaijan mostly resides with the presidency (and his parliamentary party is popular), he has little to gain from a rigged vote, and made a show of trying to rein in unreconstructed election officials and over-zealous police.
ECONOMIST: The two faces of Azerbaijan and its president | The
This Mr Aliev seems to understand that, since power in Azerbaijan mostly resides with the presidency (and his parliamentary party is popular), he has more to lose than to gain from another rigged vote, and is striving to rein in unreconstructed election officials and over-zealous police.
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