[FEATURE] Desktop app: New session should preselect the project of the session it was started from

## Problem Statement

In the Claude desktop app, with a session of some project open, pressing **New** used to open the new-session view with that project already selected in the project picker at the bottom. It no longer does — the picker comes up empty and the project has to be chosen by hand every time.

Starting another session in the project you are already working in is by far the most common reason to press New, so the default that required no action has been replaced by one that always requires an extra step and a chance to pick the wrong project.

## Proposed Solution

When **New** is pressed while a session is open, preselect that session's project in the new-session view, as it did before. Leave it empty only when New is pressed from a context that has no project.

## Alternative Solutions

Selecting the project manually each time, which is the current behaviour.

## Additional Context

This is a regression rather than a new capability: the behaviour existed and was removed.

Reproduce in the Claude desktop app: open a session that belongs to a project, press New, and observe the project picker at the bottom. Trace the New-session flow and how the current session's project is carried into it. Done means New preselects the current session's project, while contexts with no project still leave the picker empty.

Written by the indexing model from the issue text.