Why didn't my use of `target_link_libraries` produce the expected result?

Usage
,
1

Here’s the structure of my project:

root
├── CMakeLists.txt
├── lib1
│   ├── CMakeLists.txt
│   └── lib1.cpp
├── lib2
│   ├── CMakeLists.txt
│   └── lib2.cpp
└── main.cpp

This is lib2/lib2.cpp:

#include <iostream>
void lib2() {
    std::cout << "lib2 run\n";
}

This is lib2/CMakeLists.txt:

add_library(lib2 STATIC lib2.cpp)

This is lib1/lib1.cpp, lib1() calls the lib2() function.

#include <iostream>

void lib2();
void lib1() {
    std::cout << "lib1 run\n";
    lib2();
}

This is lib1/CMakeLists.txt，the scope is PRIVATE

add_library(lib1 STATIC lib1.cpp)
target_link_libraries(lib1 PRIVATE lib2)

This is the content of main.cpp,main() calls the lib1() function.

void lib1();
int main(int argc, char *argv[]) {
   lib1();
}

This is CMakeLists.txt:

cmake_minimum_required(VERSION 3.12)
project(Example)

add_subdirectory(lib2)
add_subdirectory(lib1)

add_executable(main main.cpp)
target_link_libraries(main
    PRIVATE lib1
)

My intention is: subproject lib2 creates a static library lib2, and lib1 creates a static library lib1 with lib2 linked to it and the scope used is PRIVATE, so lib2 should not be propagated to the target(main in above example) that depends on lib1. Although lib1 links to lib2, static libraries do not link to the libraries on which they depend, so main should not link to lib2. However, when I build the project and run main, the result is:

lib1 run
lib2 run

which indicates that main correctly links to lib2. Why is this the case?
I am new to CMake and have already spent a day on this problem. Thank you for taking the time.