[Leo] Previously selected `Like answer` / `Dislike answer` is cleared after relaunching the Leo

[hffvld]

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Description

Previously selected Like answer / Dislike answer is cleared after relaunching the Leo.

Steps to reproduce

1. Launch Brave
2. Open Leo
3. Ask any query and wait for the reply
4. Tap and hold on the reply > Choose Like answer or Dislike answer
5. Tap and hold the same answer again to confirm that the chosen option is shown
6. Close and reopen Leo > Tap and hold the same answer again > Observe that no option is chosen

Actual result

Previously selected Like answer / Dislike answer is cleared after relaunching the Leo since we don’t yet persist session history outside of the alive tabs.

https://github.com/brave/brave-browser/assets/128532965/88b128e8-7944-4592-a3b0-470e6ca5cbe5

Expected result

Previously selected Like answer / Dislike answer must not be cleared after relaunching the Leo.

Issue reproduces how often

Easily reproduced

Version/Channel Information:

• Can you reproduce this issue with the current Play Store version? YES
• Can you reproduce this issue with the current Play Store Beta version? YES
• Can you reproduce this issue with the current Play Store Nightly version? YES

Device details

• Install type (ARM, x86): ARM
• Device type (Phone, Tablet, Phablet): ANY
• Android version: ANY

Brave version

Brave build: 1.63.156
Chromium: 122.0.6261.43 (Official Build) (64-bit) 

Website problems only

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

Additional information

[stephendonner]

I think we'd need to first persist AI-chat history, by settings brave://flags/#brave-ai-chat-history to Enabled, but thanks for filing, and it'd be good to have this use-case documented 👍