Cargo doesn't rebuild package when changing local toolchains

Problem

Cargo doesn't detect that it needs to do a fresh build when switching from one local toolchain to another, or when updating a local toolchain, at least under some circumstances.

Steps

(I haven't verified the correctness of the steps below, but they should give an idea of the issue.)

# Build rustc.
./x.py build library/std

# Create a toolchain from it.
cp -r build/x86_64-unknown-linux-gnu/stage1/ ../toolchains/toolchain-a
rustup toolchain link toolchain-a /path/to/toolchain-a

# Make a change to rustc code.

# Build rustc.
./x.py build library/std

# Create a toolchain from it.
cp -r build/x86_64-unknown-linux-gnu/stage1/ ../toolchains/toolchain-b
rustup toolchain link toolchain-b /path/to/toolchain-b

# Build something with toolchain-a, then try to build it with toolchain-b.
cd ..
cargo new foo
cd foo
cargo +toolchain-a build

# This does not build anything.
cargo +toolchain-b build

A similar issue occurs with just one toolchain, and you want to rebuild after updating it, though this is somewhat less surprising.

# Build rustc.
./x.py build library/std

# Create a toolchain from it.
cp -r build/x86_64-unknown-linux-gnu/stage1/ ../toolchains/toolchain-a
rustup toolchain link toolchain-a /path/to/toolchain-a

# Build something with toolchain-a.
cd ..
cargo new foo
cd foo
cargo +toolchain-a build

cd -

# Make a change to rustc code.

# Build rustc.
./x.py build library/std

# Update toolchain.
cp -r build/x86_64-unknown-linux-gnu/stage1/ ../toolchains/toolchain-a

cd -

# This does not build anything.
cargo +toolchain-a build

Possible Solution(s)

No response

Notes

No response

Version

cargo 1.55.0 (32da73ab1 2021-08-23)
release: 1.55.0
commit-hash: 32da73ab19417aa89686e1d85c1440b72fdf877d
commit-date: 2021-08-23

Start by reproducing the two command sequences using ./x.py build library/std, linked rustup toolchains, and cargo +toolchain-a/toolchain-b build. Trace Cargo's build invalidation behavior when switching between or updating local toolchains. Done means Cargo rebuilds the package when the selected local toolchain has changed.