the setup:

• create a common block that will be used in different pages A,B,C....
• make a call to that block with: {{ render_esi( controller( "<controller>", ....

result:
the calls have page specific url info in the headers, and so a common block used in page A, B, C.... will be generated multiple times, one for each page. so, the cached blocks are not shared among all pages, but depend on the current URL.

The semanticPathinfo parameter prevents the blocks to be shared, every page has its own cached block. making the controller dump the the $response->headers will not show the semanticPathinfo and there is no way to change/customize that paramenter.