Wandsworth. It sounds like a punchline. It has a big roundabout with an absurd modernist squiggle atop of it. It has Battersea Power Station, lately being converted into investment receptacles for the better-off among the super-rich. It encloses Nine Elms, the posh new annexe of Shanghai springing up west of Vauxhall, defined by the culture-free St George Wharf tower. The new US Embassy is here too, which renowned architecture critic DJ Trump called: "horrible" and "in a lousy location".

Trump was right, as usual. It’s a squat glass box, pasted-over with the sort of plastic doilies you leave at the bottom of the Amazon bag, hunkering in a deliberately inaccessible island, amid the 24/7 whine of traffic. Wandsworth encloses Balham, home of the estate agent. And Clapham Common, home of the human resources consultant. City boys commute from here. The Northern Line will take you straight to Bank. Away from the river, hidden just behind the Be @ Ones, the Slug & Lettuces and the Ping Pongs, you’ll find vast avenues of 1960s estates, purpose-built dumping grounds, in suburbs that barely have names. Not dreadful, no, but dreadfully anonymous.

Wandsworth sounds like a punchline, but come the 3rd of May, its joke could become the payoff line to an era. If Wandsworth falls to Labour this week, then Corbyn wins in 2022.

Wandsworth is the crown jewel of London Toryism. It was, legend had it, Mrs Thatcher’s favourite council. They pioneered outsourcing here. The first tranche of council houses to be flogged off under Right To Buy were in Wandsworth. For years, it was a national laboratory, where boffin councillors would come up with increasingly ingenious ways to shave tuppence off the public purse. It worked. Wandsworth has the second-lowest council tax in Britain. Only Westminster – just across the water – is lower. At £723 a year (band D), it’s a good third lower than your average London borough (around £1,200), which explains why the place has been in Conservative hands for 40 years now.

Why would such a blue fortress ever change hands? Because London is changing colour. In 2008, Boris Johnson relied on the so-called "doughnut strategy" to win his first mayoral election, focusing on things – like abolishing the Western Congestion Charge, for instance – that appealed to the peripheral voters, the middle class, middle-aged folk in the comfortable burbs.

Photo: Mick Sinclair / Alamy Stock Photo

That constituency just doesn’t exist anymore. Massive population shifts and volcanic discontent over housing meant that, at 2017, London became Corbyn’s biggest win, even as he lost ground in former Labour heartlands like Yorkshire. In Tottenham, David Lammy was re-elected with 82 percent of the vote – more than Putin, as he later boasted. Directly opposite Wandsworth, Kensington (average income: £120,000) fell to Momentum type Emma Dent Coad. Within Wandsworth, Tory treasury minister Jane Ellison was kicked out, with a 10 percent swing to Labour. Pre-2017, the highest ever Labour vote share in London was 49 percent. Now, it‘s at 54.5 percent. Corbynism’s heartland is the capital.

"But everyone’s trying to play down the importance," a Momentum organiser in the borough tells me. "The Tories are in expectation management mode – if they run to only small losses, they will claim it as a victory. And we don’t want to get too cocky either."

At present, it looks like Labour will fall just short. The polls slumped back last week. Labour have 19 of 60 seats. Right now, the Tories have 39. A predicted swing of 5.35 percent to Labour won’t be quite enough to reverse that.

Mind you, there's reason to suspect that the polls could be way off. As it’s a council election, not a national one, resident Europeans get their first chance to vote since the EU Referendum. Traditionally, only a third of eligible voters bother to turn up at local elections. EU citizens might see things very differently.

"Brexit is certainly something we’re campaigning on," says the Lib Dems’ membership secretary, Thomas Arms. "There are maybe 15,000 eligible European voters in the borough. Out of a total votership of around 100,000." That’s all the swing Labour would need. Of course, the Lib Dems would love to get their votes, but the Lib Dems are nowhere here. Scan the numbers ward-by-ward, and Wandsworth is the kind of place that elects either three Labour councillors or three Tory ones. It is, as features writers love to point out, ‘a"place of contrasts".

In what may prove to be a fatal error, Labour have decided to fight the Tories on their own terrain. The Conservative strategy is clear: council tax, council tax, council tax. All their leaflets have to do is contrast your £700 bill with the £1,300 a year you’d shell out in neighbouring Lambeth. Markets in action; direct democracy at its best. Though they didn’t feel like discussing it with VICE ("Hi there. Forgive my ignorance but I’ve not heard of VICE.")

To win, Labour have promised to match them, and not to raise rates. The Lib Dems at least favour a £10 surcharge for social care. "Give them their due," Arms says, "the Tories have been very good at managing money." The situation is complicated by just how good. While they’ve been slashing services, the borough has accrued half a billion pounds, squirrelled away for some undetermined rainy day. Labour don’t seem to have a clear position on what they’d do with the windfall. "Spend it?" is the implication.

Wandsworth is dirty, residents say. The council boffins removed bins to save the money of emptying them. The results were predictable. Its child services have been rated "inadequate". The limits of chipping away could soon be breached. For the first time since 2006, a majority of those polled in the UK would be happy to see taxes rise.

Now, the 25,000 homes sold off under "Right To Buy" are no longer owned by redoubtable working class matriarchs who back Maggie. They’re the private property of landlords who don’t live in the borough, squeezing their HMO human warehouse stock, tenant-managed by eyeless agencies, until they give up on life. "I feel like we’ve been getting a lot of support from the richer wards," our Momentum organiser says. "These are people on comfortable salaries, but they’re not even owners now – they’ve dragged themselves onto shared-ownership schemes where they have a 25 percent stake in some new development. But even on these fantastically expensive developments the build quality is shoddy, there are gaps between wall and window!"