### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

When using LiteLLM (v1.86.1) as a proxy with a model configured using the **hosted_vllm** prefix, the Claude Desktop (Cowork) fails to recognize the end of the response. Although the model completes its full response and all text is displayed correctly, the UI continues to display the "Writing..." status and the timer keeps running indefinitely.

### Steps to Reproduce

<pre> [config.yaml] model_list: - model_name: qwen3.5-122b-a10b litellm_params: model: hosted_vllm/qwen3.5-122b-a10b-fp8 api_base: http://localhost:8000/v1 model_info: input_cost_per_token: 0 output_cost_per_token: 0 supports_function_calling: true supports_vision: true </pre>

1. Set up LiteLLM Proxy with the configuration above. 2. Use a Claude Desktop to connect to the LiteLLM proxy. 3. Send a prompt to the model. 4. Wait for the model to finish generating the text.

### Relevant log output

<img width="756" height="221" alt="Image" src="https://github.com/user-attachments/assets/c787b44d-38c5-41ec-a723-297ad130ba…