Euro zone bond yields rise as hopes fade for Iran breakthrough 

* Investors price in three ECB rate hikes as oil prices surge

* ECB's Joachim Nagel ​warns rates may ⁠rise if inflation expectations de-anchor

* German, Italian, French bond yields climb; ​U.S. and UK yields also rise (Updates with U.S. inflation data, refreshes prices)

LONDON, May 12 (Reuters) - Euro zone bonds sold off for a fourth day on Tuesday, as hopes faded ​for ‌a peace deal in the Iran war, which pushed up the oil price again and prompted investors to price in three rate hikes by the European Central Bank this ⁠year.

U.S. President Donald Trump said the ceasefire with Iran was "on life support" ⁠after Tehran rejected a U.S. proposal to end the conflict ​and stuck to a list of demands that Trump has described as "garbage".

"Clearly there has been a lot of back and forth gyrations that we've had over the last few weeks when it seemed there was no hope, then there was a lot of hope, then we're back to little hope. So ​it's very ‌hard to know what the final endpoint is," Sandra Horsfield, economist at Investec, said.

Germany's 2-year yield, which is regarded as more sensitive to rate expectations, was 6.4 basis points higher at 2.709%, up for a fourth day in a row as prices fell.

Interest rate expectations picked up again on Tuesday, with money markets pricing in around three 25-basis-point rate hikes from the ECB by the end of the year. The probability of a ​policy increase in June was last close to 90%.

ECB policymaker Joachim Nagel said on Tuesday the ECB must raise interest rates if the oil shock ‌resulting from the Iran war threatens to unmoor inflation expectations in the euro zone.

"Should the effects prove large or persistent, and especially if they threaten to de-anchor long-term inflation expectations, our mandate requires us to ‌act," Nagel, the head of Germany's Bundesbank, told a central bank event.

Andrzej Szczepaniak, senior European economist at Nomura, said it was almost guaranteed that inflation expectations would rise further, opening the door for the ECB to raise rates at its June meeting.

The central bank will try to get ahead ​of so-called second-round inflation effects like a wage-price spiral, he said.

"Basically, the ECB will be saying by raising rates once or twice, hey, look, we're not going to leave inflation ‌or inflation expectations unchecked. We're going to make sure we get ahead of things and ensure that inflation stabilizes at target over the medium term," Szczepaniak said.

BOND YIELDS SURGE ACROSS EURO ZONE The yield on the German 10-year bond, the benchmark for the euro zone, was 5.9 bps higher at 3.103%. Italy's ⁠10-year bond yield ⁠jumped nearly 10 bps to 3.881% and French 10-year bond yields rose 8 bps to 3.742%.

The ‌German federal statistics office on Tuesday said that EU-harmonised inflation stood at 2.9% in April, confirming preliminary figures. U.S. data showed consumer prices rose at a brisk pace for a second month ​in a row in April, as ​the war with Iran pushed up energy costs and food prices surged. U.S. Treasury yields rose 4.5 bps ‌to 4.457%.

Meanwhile, UK gilts were battered by political turmoil in Britain, where Prime Minister Keir Starmer vowed to stay in his job, defying mounting pressure within his own party. Thirty-year gilt yields hit their highest since 1998 at one point on Tuesday and were last 10 bps higher on the day at 5.77%.