The respected pollster said the reason the Foreign Secretary is popular with Conservative members may also be why she suffers against Labour in a future ballot of the country. He highlighted Ms Truss was popular with Tory members because she was playing to their ideological beliefs.
Sir John explained it meant she was less likely to be implementing policies in Government that would appeal to swing voters who will determine the outcome of the next election.
“Truss is winning this contest by appealing to the right of her party”, he wrote in The Times.
“However, they will expect her to deliver on her policy agenda, and that may make it more difficult for her to win the crucial centre ground when a general election day does come.”
Mr Sunak has found himself repeatedly more popular among the country as a whole, despite falling behind among his own party members.
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Sir John added: “One of the former Chancellor’s biggest problems is that on almost every issue he is winning the support of those who back the less popular position among Tory members.”
His analysis comes after a YouGov poll last night found Ms Truss was the preferred candidate among the vast majority of party members.
In total 69 percent of those surveyed said they would back the Foreign Secretary, compared to 31 percent who said they would back her rival.
It means Ms Truss’s lead of Mr Sunak has grown from 24 points to 38 points when compared to YouGov’s last survey on July 20-21.
At a leadership hustings last week, the South West Norfolk MP vowed that she would continue to be Conservative in her approach as Prime Minister.
She said that the party’s historic election victory in the 2019 election was proof it was what voters wanted.
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Ms Truss told party members: “We need to be proud to be Conservatives.
“I’m going to campaign as a Conservative, and I will govern as a Conservative.
“The people who voted for us in 2019 in seats like Keighly, in seats like Dewsbury, they didn’t vote for us because they wanted Labour policies.
“They were fed up of the Labour Party, they were fed up of the failing Labour councils.
“They voted for us because they wanted opportunity, because they aspired to better things, because they wanted enterprise in their area.
“And that is what I, as your prime minister, am determined to deliver.”
Responding to today’s leadership poll, Ms Truss said she was “not taking anything for granted”.
“There is still a long way to go in this leadership election,” Ms Truss said when asked if she was looking forward to being Prime Minister.
“What I’m about is unleashing opportunity across the United Kingdom, getting our economy growing and delivering for people all over the country.”