My initial fears about how the EU referendum would play out appear, unfortunately, to have been realised. The debate over the size of the economic costs has been turned into Project Fear by Brexit campaigners (with the media’s help), leaving most voters to believe they would be no worse off if we left. Media coverage has been dominated by Conservative politicians and political commentators, rather than those with some expertise who might have been able to convince the public that the costs of Brexit were not just another macroeconomic forecast. Among all the tedious noise of claim and counter claim, one apparent fact stands out: we cannot control EU immigration from within the EU. How can Conservative leaders who have pandered to popular concerns about immigration with impossible targets now convincingly turn around and say immigration isn’t really so important? To do so would lack credibility, so they have not even tried.
Just as in the Scottish independence referendum, it is going to be up to those who are not Conservative politicians to save the day for Remain. One way to do that would be to take the immigration issue and play on the general public distrust of politicians.  Ask this: if we Leave, why do you think those in charge will really cut immigration from the EU? After all, net immigration into the UK from outside the EU is at least as high as from the EU, and the UK has complete control over non-EU immigration. Yet non-EU immigration has hardly fallen since 2010, and is itself way above the government’s own target.
There is a simple reason that Conservative politicians have not brought non-EU immigration down, and that is because the costs to the economy of doing so clearly outweigh the benefits. They continue to pretend they are trying to bring immigration down (and doing some harm in the process) only because immigration is a convenient scapegoat for the impact of their own policies.
If we cut immigration, it would not ease pressure on public services, because on average migrants - because they are young - pay more in taxes and utilise public services less than non-immigrants. In fact the reverse would happen: a larger government deficit would see more cuts to the NHS and to tax credits. As Conservative MP Dr Sarah Wollaston, who defected from Leave to Remain, said: “If you meet a migrant in the NHS, they are more likely to be treating you than ahead of you in the queue.” Yes of course we could train more doctors and nurses, but the politicians in charge of Leave and who would be running things after Brexit have been reducing the share of national income spent on the NHS.
Cutting immigration might in itself directly improve the pay of low earners if the numbers of jobs remained the same. But the number of jobs would not remain the same. Both UK and overseas companies would take their jobs abroad so that these companies benefited from being inside the single markets. The net outcome for British workers would almost certainly be worse. 
Leaving the EU does not give UK voters control over their border and who comes in. It gives control to the politicians running the Leave campaign. Politicians who have so far cut spending on the Border Force budget. In particular do you trust Boris Johnson, whose main interest in supporting Brexit is that it will let him rather than George Osborne be the next Prime Minister, and who has argued in the past that low immigration could lead to economic stagnation?
This referendum is about trust. Do you trust 9 out of 10 economists who say that Brexit will be bad for the economy, or do you trust the politicians who say they will cut immigration if you Vote Leave, but have failed to significantly reduce immigration from outside the EU over the last six years?
 This line of attack is partly suggested by this post by Jonathan Portes, but also by the finding here that although immigration is the main issue for those voting leave, they are somewhat divided on how much Brexit will solve the ‘migration problem’. See also this and this. A worthy but ineffectual alternative is to stress the benefits of immigration head on. Desirable though that might be in the longer term I don’t think you can turn around the ‘received wisdom’ in 10 days. An unwise alternative is to emulate what the Conservatives did, which is to say that we need to convince the EU that migration needs to be controlled more strongly than it is at present.
 I suspect arguing about the size of the impact of immigration on low pay (is it large or small) will achieve little, particularly as reasonable opinions differ. It is a bit like arguing over the £350 million a week figure: it plays into Leave’s hands by focusing on the negative (there is a negative effect of immigration on low pay, and there is a net contribution to the EU).