client's subsequent connection not with context path after transport

[yingchousun]

Inspector Version
Inspector 0.19.0

• [e.g. 0.16.5)

Describe the bug
I use spring ai wirte a simple mcp server, its context path is /xxx。use Inspector 0.19.0，i config the sse url is http://127.0.0.1:8080/xxx/sse, proxy connection，then the first connection is ok, but then send message is 404.

if i start my spring boot app without context path, all is ok.

To Reproduce
Steps to reproduce the behavior:

1. Go to '...'
2. Click on '....'
3. Scroll down to '....'
4. See error

Expected behavior
Inspector client should recognize the context path, and add it to all sequential access

Screenshots
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Environment (please complete the following information):

• OS: [e.g. iOS]
• Browser [e.g. chrome, safari]

Additional context
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Version Consideration

Inspector V2 is under development to address architectural and UX improvements. During this time, V1 contributions should focus on bug fixes and MCP spec compliance. See CONTRIBUTING.md for more details.