`format_batch_tag` handed out `set N` ordinals from one process-wide table,
so every conversation on a shared backend and every child's nested fan-out
advanced the same counter. A user's second wave of 15 lanes rendered as
`[set 20 · 13/15]`, which reads like 20 batches were spawned.
Scope the ordinal table by the parent conversation (`parent_agent.session_id`)
and thread the parent through the three render sites (batch header,
completion lines, child tree-line prefix via the shared session_ref). The
first fan-out in a conversation is `set 1`, the next `set 2`; sibling
conversations and nested child batches no longer inflate it.
Invariant test proven red on origin/main, green with the fix.
0cb996d977