Make `@`modes sticky

Currently @mode is lost when continuing conversation which risks losing features such as fetching source files and detailed diagram prompting. It would be nice if it was sticky, ie. after a prompt gets sent let the mode stay in the prompt box. For extra nice UX it could be in a pill that can be single-backspaced.

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The first part of this implementation is in the CLI / RPC. The explain response should indicate clearly what the command mode of the user message was (explicitly stated or implicitly assigned).

If Navie is not clearly indicating implicit command modes back to the caller, this should be added and used by the CLI / RPC

The second part of this implementation is in the UI Vue components, specifically in the search box in which the user types their question.

For follow up questions, the text input should be preloaded with the command prefix of the most recently asked question (whether explicit or implicit).

Start with the CLI/RPC explain response and verify whether it clearly reports the explicit or implicit command mode. Then inspect the Vue search box components and follow-up prompt flow. Done means the latest command prefix is retained in the input, with the proposed pill/backspace behavior if feasible, and explicit or implicit modes are consistently surfaced by the CLI/RPC.