Follow up of #12452 - incognito_used_timestamp is not recorded correctly

[GeetaSarvadnya]

Description

Follow up of #12452 - incognito_used_timestamp is not recorded correctly

Steps to Reproduce

1. Clean profile 1.18.68
2. Open a PT from normal window (using sidebar/hamburger menu)
3. Open brave://local-state file in the normal tab window
4. Note down the incognito_used_timestamp value
5. Close the PT
6. Open a TOR window tab from the normal window (using sidebar/hamburger menu)
7. Refresh the local-state file which is opened in step 3
8. Note down the incognito_used_timestamp value
9. Values from step 4 and step 8 are different

Actual result:

incognito_used_timestamp value is different when TOR window is opened

Expected result:

As per the description #12452 (comment), opening a TOR window shouldn't change the incognito_used_timestamp value

Reproduces how often:

Easy

Brave version (brave://version info)

Brave	1.18.68 Chromium: 87.0.4280.67 (Official Build) dev (64-bit)
Revision	0e5d92df40086cf0050c00f87b11da1b14580930-refs/branch-heads/4280@{#1441}
OS	Windows 10 OS Version 2004 (Build 19041.630)

Version/Channel Information:

• Can you reproduce this issue with the current release? Yes
• Can you reproduce this issue with the beta channel? Yes
• Can you reproduce this issue with the nightly channel? Yes

Other Additional Information:

• Does the issue resolve itself when disabling Brave Shields? NA
• Does the issue resolve itself when disabling Brave Rewards? NA
• Is the issue reproducible on the latest version of Chrome? NA

Miscellaneous Information:

cc: @brave/legacy_qa @iefremov @rebron

[GeetaSarvadnya]

Discussed with @iefremov over the call - Refresh Brave://local-state file after step 5 fixes the issue. Hence closing the issue.