decorated for older browsers https://gist.github.com/jherax/a81c8c132d09cc354a0e2cb911841ff1

isPrivateMode.js

function isPrivateMode() {
        return new Promise((resolve) => {
          const on = function(){ resolve( true ); }
          const off = function(){ resolve(false); }
          const testLocalStorage = function(){ 
            try {
              if ( localStorage.length ) {
                off();
                return;
              } 
              localStorage.x = 1;
              localStorage.removeItem('x');
              off();              
            } catch ( e ) {
              navigator.cookieEnabled ? on() : off();
            }
          };          
          // Chrome
          if ( window.webkitRequestFileSystem ) {
            return void window.webkitRequestFileSystem( 0, 0, off, on );
          }
          // FF
          if ( "MozAppearance" in document.documentElement.style ) {
            var db = indexedDB.open( "test" );
            db.onerror = on;
            db.onsuccess = off;
            return void 0;
          }
          // Safari
          if ( /constructor/i.test( window.HTMLElement ) ) {
            return testLocalStorage();
          }
          // IE10+ & Edge
          if ( !window.indexedDB && ( window.PointerEvent || window.MSPointerEvent ) ) {
            return on();
          }
          // others
          return off();
        });
      }