The debate on immigration during this EU referendum campaign cannot end up as just sirens on one side and silence on the other. Those leading the leave campaign are being deliberately dishonest to inflame public concern. Vile leave posters linking Europe with the Orlando attack are designed to inflame fear and anger. So too are the hype and lies about Turkey joining the EU.

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But those of us who believe strongly that we need to remain can’t ignore the issue. The home secretary, Theresa May, has said nothing about immigration at all in this campaign. Much as David Cameron would like to talk only about the economy and jobs, it isn’t enough. Much as Labour needs to argue strongly for staying in to protect workers’ rights, we also have to respond to concerns on immigration too.

For centuries Britain has benefited from the dynamism and hard work of those who have come here from abroad, and we need international talent and ideas. But that’s why it is so important that there is public confidence in the system – and that it is controlled and managed so it is fair.

Public concern has grown in countries across the world as international migration has risen. Responding to this is a challenge for our generation, as communities worry about the pace and scale of change. Workers, especially the low paid, worry about the impact on their jobs and pay.

Ignoring the problem doesn’t work. People get frustrated that legitimate concerns aren’t heard, and populists end up having free rein to scream, shout and exploit people’s fears. We cannot let that happen in the final days of this referendum campaign. The leave campaign needs to be challenged on its lies. Brexit needs to be exposed as a false promise. But as remain campaigners we should also show there is a sensible way forward with more reform.

Michael Gove and Boris Johnson are being deliberately dishonest and divisive. Turkey isn’t joining the EU. A million people aren’t coming; 76 million people aren’t coming. They meet one out of 35 criteria they need to join. Cyprus will veto them. So will Greece – and Britain would have a veto in future too. The NHS crisis is the fault of the Tory government, not immigration. Indeed, pulling out would hit growth and mean less money for the NHS, not more, as our economy is hit.

Leaving Europe would make it harder, not easier, to tackle the serious problem of illegal immigration. We depend on the French to keep our border checks in Calais and on Europol to stop the smuggler gangs. At the same time we need to cooperate over the Syrian refugee crisis and help those fleeing persecution and conflict. That crisis won’t go away if we pull out, and no country can respond alone.

Nor is Brexit likely to lead to big cuts in legal migration either. Over half of migration comes from outside the EU. Johnson has promised access to the single market and a good trade deal – so given Europe’s current rules, we are most likely to end up like Norway or Switzerland, with little change to immigration after all. Because if they don’t do a good trade deal, the impact on jobs and growth would be disastrous.

Michael Gove and Boris Johnson are being deliberately dishonest and divisive. Turkey isn’t joining the EU

But exposing the leave arguments isn’t enough. We also need to show that voting remain doesn’t mean voting for the status quo.

In David Cameron’s renegotiation in Europe he got the benefit rules changed so people coming to Britain have to pay in before they claim. That’s a start. But here’s five more reforms remain campaigners could sign up to now – part of a national plan to build a new sensible consensus on immigration instead of the current damaging polarisation.

1 New employment rules

To stop immigration being used to undercut local terms and conditions. That includes Jeremy Corbyn’s call for us to strengthen the posted workers directive.

2 A proper migration impact fund

As Labour MP Jo Cox has argued, EU citizens working in Britain have paid in £20bn more than they’ve taken out in public services the last 10 years. Let’s put some of that money into a fund for public services and integration in areas where migration has increased, as Gordon Brown has said.

3 Stronger border controls

Our border force needs more investment, not less, to deal with illegal migration. We need a new plan with Europol to stop the people smugglers crossing the Channel. And proper exit checks to count people in and out.

4 Reforms to free movement

When Britain takes the presidency of the EU next year we should use it to build an agreement from the inside, rather than the last-minute special pleading David Cameron tried before: to change the rules for new countries; to get the rest of Europe to change Schengen so they bring back border checks to stop people trafficking; to get all countries to do more to help refugees, but to develop new kinds of controls and brakes to manage economic migration.

No one should make false promises, nor should we be afraid of rolling up our sleeves in Europe and arguing for reform. When I called six months ago for the restoration of border controls in the Schengen zone to stop trafficking gangs, everyone said it could never happen. Now many countries are restoring border checks. And our best chance of keeping both single market access for our economy and getting sustainable immigration reform is by staying in and working for it, not by pulling out.

5 Build a new national consensus on immigration

This angry debate is damaging, whether we are in or out. For three years before the last election I went to public and community meetings on immigration all over the country, and there is more common ground than you would think.

Few people believe we should pull up the drawbridge, most think immigration should be better managed and controlled. Most people like the freedom to travel, but they want more safeguards in place. Most people think we should do more to help refugees, and that we benefit from international students, but they are angry about illegal immigration and worried about the scale of low-skilled migration affecting jobs.

There are countless hard-headed reasons for staying in Europe – jobs, public services, workers rights and co-operation which helps keep our country safe. There are inspiring reasons too – standing with our closest neighbours as we have done since the second world war, using debate and diplomacy to work out our differences instead of weapons and war.

But the danger is all these powerful reasons don’t get heard because of the immigration debate. So let’s not duck it.

If we leave this to the Brexit campaign we will not only end up outside of Europe, divided from our international allies, we will also end up with deep division across our own country as well.