Chrome Version: 56.0.2924.53
OS: Chrome

1. Sign in a primary and secondary profile
2. Have browser windows open in primary desktop
3. Switch to secondary desktop; have nothing open there
4. Ctrl+T

Expected: new browser window opens to the new tab page
Actual: nothing happens; switching to the primary desktop shows a new tab was opened in a primary profile browser window on that desktop

Ctrl+N has a similar effect.

With an app open on the secondary desktop (but not a browser window), Ctrl+T will still open a tab for the primary desktop.

Only if a tab is open on the secondary desktop will Ctrl+T or Ctrl+N open a tab on the secondary desktop.