Does not seem to honour ESI configuration

We re using Drupal 10 and a custom Boostrap 5 theme and we have tried over and over to use this LiteSpeed Cache module every time a newe version comes out but we have never succeeded with it.

We thought it was was time we put pen-to-paper so that someone wise in the community can put us right!

Her goes...

WHEN the LiteSpeed cache kicks in with our shared hosting it is absolutely spectacular and we get pages served in a fraction of a second!

BUT we have never been able to use it because it does not behave as we understand and expect.

### SYMPTOMS
For logged in contributor users, TWO things should change:

1) In the horizontal menu across the top (`id=block-bit-by-bit-horizontalmenu`), the last menu item should change to 'Me' and led to a submenu as shown below.

2) The Admin Toolbar (`id=toolbar-administration`) should show on the lefthand side as shown below.


### IT WORKS and then it DOESN'T WORK
**Sometimes this works fine** and then we switch to another tab (within the same session) where both elements should appear the same for the logged-in user but we get neither as below...


No amount of browser cache-clearing makes any difference.

### CONFIGURATION
As far as we understand it, if we want both those blocks to be considered as ESI blocks then we should configure the LSCache module as below...


### HELP and ADVICE
Can anyone please offer some help and advice as to whether we have simply misunderstood or whether we have configure the module wrongly.

The facility to be able to use the LS Cach reliable would fundamentally change the end users' experience and so we are very keen to try to get to the bottom of this.

Thank you

No source file or test is identified. Start by reproducing the Drupal 10 setup with the reported ESI configuration and logged-in contributor behavior across tabs; done means the horizontal menu and Admin Toolbar reliably appear for that user without relying on browser cache clearing.

Written by the indexing model from the issue text.