Tories said at the weekend that they hoped to humiliate the SNP by defeating a number of their leading MPs, including Salmond.
And although the SNP will certainly "win" the general election in Scotland, just as they have the local government elections, they now have cause for concern.
SNP MSP Derek Mackay, who has been announced as the party's campaign director for next month's general election, said the council election results would be an "excellent springboard".
The share of the SNP's first preference vote is actually exactly the same as five years ago, when the last round of these elections was held.
Ms Sturgeon started the first full week of campaigning at Inveralmond Brewery in Perth with local SNP MP Pete Wishart, where she practised pulling pints.
Ms Sturgeon said: "The issue at the heart of this election, not just in Perth and Perthshire but across Scotland, is a very straightforward one".
Ms Davidson, the Scottish Tory leader, has already declared her party has a "pretty good" chance of winning the Moray constituency. "It's time to focus on this crucial general election campaign - and we can take absolutely nothing for granted", May will tell the event on Monday.
The full results of last week's Scottish council election have been published - showing the SNP won 32% of the first preference votes.
Ms Davidson said: "I think you have seen in both language and design, the SNP attempt to assimilate the party for Scotland and Scotland for the party".
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Labour candidate Callum O' Dwyer added: "A vote for Labour is the only way to replace this failing Tory government and stand against a second divisive independence referendum".
Nicola Sturgeon has dismissed claims that the surge in Tory support in Scotland could derail her plans for a second independence referendum as "ludicrous".
Elsewhere in the west of Scotland the party's vote also increased especially sharply in West Dunbartonshire (up ten points), and in Inverclyde (up seven).
"It is only by working flat-out, every day, from now to 8 June that we can gain the trust of the British people and earn their support on polling day".
"We are determined to speak up for people in Scotland who are beginning to find their voice".
But a resurgent Tory party have said the seat is a key target and have urged Unionist voters to back them. They seem to want to silence Scotland's democratic voice.
"In terms of the strengthening of Theresa May, certainly if you look at the results in England, Theresa May is on course for a landslide victory - that raises a really important question in Scotland over the next few weeks".
The Scottish Conservative leader, Ruth Davidson, has told supporters that there are no longer any "no-go areas" for her party, as she tried to establish the Tories as the main opposition to the Scottish National Party.
Labour also had a higher percentage of first preference votes than the SNP in Coatbridge, Chryston and Bellshill, up 1.65% percentage points, and was 0.91 percentage points higher in Lanark and Hamilton East.