feat: correlate inference requests with session ids

[chenghao-mou]

Inference gateway calls need a correlation key that follows each transport lifecycle. Session IDs are SDK-owned.

In-job correlation:

Component	Session header (key: value)	Session scope	Per-request ID pattern
STT	X-LiveKit-Session-ID: inference_<hex12>	WebSocket	Service-defined; fallback: session ID
TTS	X-LiveKit-Session-ID: inference_<hex12>	Pooled WebSocket	inference_<hex12>_tts_<hex12> per synthesis
LLM	X-LiveKit-Session-ID: inference_<hex12>	HTTP attempt	Service-defined; no derived SDK ID
EOT	X-LiveKit-Session-ID: inference_<hex12>	WebSocket	inference_<hex12>_eot_<hex12> per prediction
Barge-in	X-LiveKit-Session-ID: inference_<hex12>	WebSocket	created_at nanoseconds per frame; no string ID

Outside a job, no session header is sent and existing request-ID formats remain. Custom headers cannot replace the SDK-owned session ID.

Addresses LKINF-344

Initial prompt and agent context

Model: GPT-5.6

we need to pass the client side request id to inference as a session id header $grill-me

I am not asking about job, I am talking about inference stt/tts/llm/eot/bargein

we should change this a bit to make it semantic:

• connection session id: generated once, use for the connection header (stt, tts, and llm http requests)
• tts per request: reuse the session_id but append a suffix so as to correlate locally.

tts or eot

inference_ is enough

[devin-ai-integration]

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