In the Shin-Okubo neighborhood of Tokyo, smells of Korean food and snatches of the language waft in the air. A supermarket selling kimchi sits next to an Indian-run kebab shop — the latter complete with leaflets promoting Islam, the religion of the Calcutta-born owner. A local real estate agent advertises staff that speak Chinese, Vietnamese and Thai alongside the floor plans for tiny Tokyo apartments.

Shin-Okubo is a rarity in Japan. The country has remained relatively closed to foreigners, who make up only 2 percent of the population of 127 million, compared with an average of 12 percent in the Organization for Economic Cooperation and Development, a club of mostly rich countries. Yet Japan is especially short of workers. Fully 83 percent of firms have trouble hiring, according to recruiting firm Manpower, the highest of any country it surveys. And the squeeze is likely to become much worse. The population is projected to drop to 87 million by 2060, and the working-age population (15-64) from 78 million to 44 million, because of aging. The Keidanren, the Japan Business Federation, and prominent business leaders such as Takeshi Niinami, the head of Suntory, a drinks company, have long called for more immigration.

Prime Minister Shinzo Abe says he would prefer to raise the relatively low proportion of Japanese women who work, and to keep all Japanese working later in life, before admitting droves of foreigners. But his government has nonetheless taken a few steps to boost immigration. It has quietly eased Japan's near-ban on visas for low-skilled workers, with agreements to allow foreign maids to work in special economic zones. It is now talking about relaxing requirements for Filipino caregivers.

The authorities have also made student and trainee visas easier to obtain and have turned a blind eye to those who exploit them to recruit staff for jobs that involve very little study or training at kombinis (the ubiquitous corner stores, often staffed by Chinese) or in forestry, fishing, farming and food-processing. It may extend trainee visas from three years to five. Abe has also boasted that he will reduce the time nonpermanent residents need to live in Japan before becoming eligible for permanent residence to the "shortest in the world" — probably to less than three years (far from the shortest) from the current five.

All this is starting to make a difference. The number of foreign permanent residents reached a record 2.23 million last year, a 72 percent increase over 20 years ago — and the number of people on nonpermanent visas is also rising. But the goal seems to be a surreptitious increase in the number of temporary workers and a more accommodating system for skilled workers, not the settlement of foreigners. Only tiny numbers of foreigners become Japanese citizens and even fewer are granted asylum: only 27 in 2015, a mere 0.4 percent of applicants.

A few voices advocate opening the door more widely. Hidenori Sakanaka, a former immigration chief who now leads the Japan Immigration Policy Institute, a think tank, reckons Japan needs 10 million migrants in the next 50 years.

At the very least, the country needs a clear policy on bringing in menial foreign workers, rather than ignoring the abuse of student and trainee visas, says Shigeru Ishiba, a prominent lawmaker in the Liberal Democratic Party who is expected to challenge Abe for the party's leadership in 2018.

Public opinion seems to be gradually shifting. The authors of a recent poll by WinGallup were surprised that more Japanese favored immigration than were against it — 22 percent to 15 percent — although a whopping 63 percent said they were not sure.

A warm embrace for lots of foreigners is unlikely. The country prides itself on its homogeneity, and although the media no longer reflexively blame foreigners for all social ills, discrimination is still rife.