British workers haven’t been losing out to foreign labour and are unlikely to see job prospects dramatically improve because of Brexit immigration cuts, according to a survey of UK companies.

While almost half of firms faced skill or labour shortages in the past 12 months, less than 10 per cent said they targeted overseas workers to fill the gap, instead investing more in recruitment and training, the British Chambers of Commerce said Wednesday.

It added that the results “challenge the myth that UK firms are ignoring local workers in favour of overseas labour.”

There was also a note of caution for those who say that reducing the inflow of European Union workers as part of Brexit will help jobs prospects for Britons. Just one-in-five UK companies said a restriction on access to EU labour would prompt them to focus recruitment on local employees.

The free movement of labour featured through the campaign before last year’s EU referendum, playing on voters’ concerns that growing numbers of workers from overseas are snatching jobs and pushing down wages. It’s also played a part since then, with the government signaling it’s willing to prioritize migration controls over access to the single market.

Forty per cent of UK businesses have employees from other EU countries, while 23 per cent have workers from outside the bloc, the BCC survey showed. Almost 40 per cent say future restrictions on EU workers would have a negative impact on their business.

The risk of a shortage of workers is increasing as UK unemployment remains at historically low levels. The jobless rate stayed at an almost 40-year low of 4.4 per cent in the three months through July, economists said in a survey before data due later on Wednesday.