Invoking Shared Common Module Flow from Multiple Architect Inbound Flows via HTTP

Could someone explain the correct HTTP payload structure for invoking a shared flow (Common Module) from multiple distinct inbound call flows using the Architect Data Action API?

I am attempting to modularize our IVR logic by extracting common authentication steps into a reusable flow. However, when I trigger the shared flow via a standard HTTP POST to /api/v2/architect/flows/{flowId}/executions, the execution context does not seem to propagate the necessary conversation metadata correctly, resulting in a silent failure where the shared flow starts but lacks access to the original call’s DTMF inputs.

Here is the JSON payload I am currently sending:

{
 "flowId": "shared-auth-module-id",
 "context": {
 "conversationId": "{{conversation.id}}",
 "dtmfSequence": "{{dtmf.sequence}}"
 }
}

The API returns a 200 OK, but the shared flow logs show undefined variables for the DTMF sequence. I suspect the context object mapping is incorrect or that I am missing a specific header required to bind the execution to the parent conversation’s state.

Is there a specific way to map the parent flow’s variables into the child flow’s execution context using the REST API, or should I be using a different mechanism entirely? I want to avoid duplicating the authentication logic across ten different inbound flows.