fix(tui3): keep Home tasks in the displayed project

[santoshkumarradha]

A /task typed on Home in project B could run in project A after the command cleared its draft. Clearing rebuilt Home’s rows and moved the cursor to A’s newer conversation before the target was read. Ordinary messages read the target earlier. An engine/workspace mismatch check cannot catch this because the engine correctly answers for the already-wrong target.

Capture the displayed target before consuming the slash draft, then pass it through the same conversation-opening implementation used by ordinary messages. A keyboard regression reproduces the task-only mismatch on current audit PR #1604 (80aff6b9f); the plain-message control stays in B. Independent code review found no blocking issue.

Validation on Spark at exact source/runtime 3cbffa975:

• Focused Home regressions passed, the light gate passed, and the complete affected internal/tui3 package passed in two shards (141 seconds). Required GitHub CI is fully green.
• Real hosted tmux workflow kept A’s engine alive after its window closed, with an unpublished README edit. Home in B then started an actual task in B. It created the useful SHOPPING.md on B’s retained task branch at commit 611bc7e; only that report changed. B’s main branch stayed clean, and A’s tracked files, refs and unpublished edit remained identical.
• A plain message submitted from Home opened a fresh B conversation, read the actual committed report and ran the pantry CLI and 19 tests. An independent test rerun passed. Both isolated fixture engines were stopped by exact workspace after they became idle.
• All 48 recorded calls used OpenRouter deepseek/deepseek-v4.1-flash, including auxiliary roles: 12 calls for A setup and 36 for B’s task/control. This used ordinary hosted chat with ask approvals and --one-model; Model Pool was disabled as the disclosed workaround for separately fixed fix(chat): defer independent pool judges in one-model runs #1610. It does not claim alternate model, remote-host or nondefault project-picker acceptance.

Audited screenshots, GIF/MP4, both original terminal recordings, binary identity, model receipts and project-preservation proof are staged locally for the coordinator’s direct attachment. They are not yet public media links; no evidence commits were created for this batch.

This is a separate batch2 fix. It is not integrated into Santosh/dev or aggregate #1627, and no shared timer was changed.

Fixes #1466.

[santoshkumarradha]

Superseded by the Santosh-authored consolidation #1632. The exact source revision and workflow recordings are included there. The aggregate remains draft pending final combined validation; this change has not yet merged into dev. Source history is preserved.