http://www.theverge.com/2015/7/6/8899807/instagram-1080p-pictures-photo-upload
The Verge recently made known the latest app versions of Instagram are uploading photo
As of 26th March 2018, Instagram changed their links. Here's a working solution to get 1080x1080 resolution images from API.
Get the user_Id from https://www.instagram.com/{username}/?__a=1
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Paste the user_id in this link https://i.instagram.com/api/v1/users/{user_id}/info/
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Get the hd url link from hd_profile_pic_url_info
. :)