fix(ai): disable deep research during active runs

[sdolidze]

Closes: https://linear.app/lightdash/issue/PROD-9372/disable-deep-research-while-another-run-is-active-in-the-thread

Summary

Disables the Deep research composer control while the current thread has a starting, queued, or running research run. Regular chat stays available, and the control returns when the run completes, partially completes, fails, or is cancelled.

Root cause

The one-click control owned only local composer mode. Although the Deep research registry and server list already tracked thread-scoped run lifecycles, the composer never consumed that state.

<DeepResearchModeControl
    mode={composerMode}
    onModeChange={setComposerMode}
/>

PR #26589 introduced the direct toggle and left same-thread concurrency feedback for this follow-up. Without active-run state, the control stayed interactive throughout an existing run.

Fix

Derives one active state from optimistic registrations and persisted non-terminal runs, then applies it only to the Deep research path.

• useDeepResearch.ts reconciles optimistic and server runs by UUID and polls while a run remains non-terminal.
• AgentChatInput.tsx disables duplicate research starts, resets active composers to Ask, and keeps normal messages available.
• DeepResearchModeControl.tsx exposes an actionless explanation on hover and keyboard focus for both control variants.
• Backend execution, API contracts, and database state remain unchanged.

Before

same-thread run active
        |
        v
Deep research toggle enabled ---> second start remains selectable

After

same-thread run active
        |
        +--> Deep research disabled + hover/focus explanation
        |
        +--> Ask composer enabled ------> regular message sends

terminal run state ---------------------> Deep research re-enabled

Test plan

• Focused Vitest: 3 files, 21 tests pass, including the original failing regression.
• Full frontend Vitest: 241 files, 1,903 tests pass.
• pnpm -F frontend typecheck
• pnpm -F frontend lint — zero errors; unrelated existing warnings only.
• pnpm -F frontend format
• Verify optimistic and persisted active states, terminal states, thread isolation, no-thread behavior, regular Send, and actionless hover/focus explanation.

[linear-code]

PROD-9372

[github-actions]

🧪 Test Selection

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Test	How to trigger manually
Backend API Tests	Add test-backend to PR description
Timezone Tests	Add test-timezone to PR description
CLI Tests	Add test-cli to PR description

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[github-actions]

Your preview environment pr-26636 has been deployed.

Preview environment endpoints are available at:

• https://lightdash-pr-26636.lightdash.okteto.dev
• https://lightdash-preview-pr-26636.lightdash.okteto.dev

[github-actions]

Preview Environment

🌐 URL: https://lightdash-preview-pr-26636.lightdash.okteto.dev

📋 Logs: View in GCP Console

🔧 SSH: ./scripts/okteto-ssh.sh 26636

[lightdash-bot]

🎉 This PR is included in version 1.55.0 🎉

The release is available on:

• 1.55.0
• GitHub release

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