Failed to execute `rustup update` in the Rust `devcontainer` (due to OverlayFS restrictions?)

I get the same problem calling rustup update in a devcontainer. The devcontainer just installs 1.84 and I need 1.85. I get

vscode ➜ /workspaces/my_project $ rustup update
info: syncing channel updates for 'stable-x86_64-unknown-linux-gnu'
info: latest update on 2025-02-20, rust version 1.85.0 (4d91de4e4 2025-02-17)
info: downloading component 'clippy'
info: downloading component 'rustfmt'
info: downloading component 'rust-src'
info: downloading component 'rust-analysis'
info: downloading component 'rls'
info: downloading component 'cargo'
info: downloading component 'rust-std'
info: downloading component 'rustc'
 69.5 MiB /  69.5 MiB (100 %)  27.2 MiB/s in  2s ETA:  0s
info: removing previous version of component 'clippy'
info: rolling back changes
error: could not rename component file from '/usr/local/rustup/toolchains/stable-x86_64-unknown-linux-gnu/share/doc/clippy' to '/usr/local/rustup/tmp/58z480gzx9kr9vug_dir/bk'
info: checking for self-update

  stable-x86_64-unknown-linux-gnu update failed - rustc 1.84.1 (e71f9a9a9 2025-01-27)

info: cleaning up downloads & tmp directories

Originally posted by @Psy-Kai in #4181

The issue names no repository files or tests. First reproduce rustup update in the reported Rust devcontainer and investigate the failed component rename under OverlayFS; done means the toolchain updates successfully without rollback in that environment.