[OpenTelemetry Instrumentation] Rush update failing in nightly runs

[v-jiaodi]

OpenTelemetry Instrumentation nightly test runs are failing with:

Error message:
.
└─┬ @rush-temp/azure-opentelemetry-instrumentation-azure-sdk-test
├─┬ @opentelemetry/core
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
├─┬ @opentelemetry/sdk-trace-base
│ ├── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
│ ├─┬ @opentelemetry/core
│ │ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
│ └─┬ @opentelemetry/resources
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
└─┬ @opentelemetry/sdk-trace-node
├── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
├─┬ @opentelemetry/context-async-hooks
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
├─┬ @opentelemetry/propagator-b3
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
└─┬ @opentelemetry/propagator-jaeger
└── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
The command failed:
/mnt/vss/_work/1/s/common/temp/pnpm-local/node_modules/.bin/pnpm install --store /mnt/vss/_work/1/s/common/temp/pnpm-store --no-prefer-frozen-lockfile --strict-peer-dependencies
ERROR: Error: The command failed with exit code 1
Giving up after 3 attempts
Deleting the "pnpm-store" folder
ERROR: The command failed with exit code 1

For more details check here:

• https://dev.azure.com/azure-sdk/internal/_build/results?buildId=1798669&view=results

[jeremymeng]

Now this is blocking the rush update --full automation
https://dev.azure.com/azure-sdk/internal/_build/results?buildId=1832991&view=logs&j=ced21a9a-dda2-553c-e1fe-5294ad6c26a1&t=054ca053-fc29-50ab-055c-0b4011aca4ac (MS INTERNAL)
.
└─┬ @rush-temp/monitor-opentelemetry-exporter
├─┬ @opentelemetry/instrumentation-http
│ └─┬ @opentelemetry/core
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
└─┬ @opentelemetry/sdk-metrics-base
├─┬ @opentelemetry/core
│ └── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0
└─┬ @opentelemetry/resources
└── ✕ unmet peer @opentelemetry/api@">=1.0.0 <1.2.0": found 1.2.0

[mpodwysocki]

@jeremymeng this PR should fix it #23115