diff --git a/.github/workflows/Downgrade.yml b/.github/workflows/Downgrade.yml
index 1ac81a1..fe253d3 100644
--- a/.github/workflows/Downgrade.yml
+++ b/.github/workflows/Downgrade.yml
@@ -52,17 +52,18 @@ jobs:
       - uses: julia-actions/cache@v2
       - uses: julia-actions/julia-downgrade-compat@v2
         with:
-          skip: Test
+          skip: Test,UUIDs,Libdl
           projects: ., test
           mode: forcedeps
-      - uses: julia-actions/julia-buildpkg@v1
-        env:
-          PYTHON: ""
-      - name: Run tests without coverage
-        uses: julia-actions/julia-runtest@v1
-        with:
-          coverage: false
-          allow_reresolve: false
-          force_latest_compatible_version: false
-        env:
-          PYTHON: ""
+      # We run the tests manually instead of using julia-action/julia-builpkg and julia-action/julia-runtest or `Pkg.test`
+      # because otherwise the downgraded Manifest.toml is not used in the tests under julia <v1.12
+      # see also https://github.com/julia-actions/julia-downgrade-compat/blob/00f940b7be2b50389571ed016d603be561649103/README.md#L89
+      - name: Run tests
+        run: |
+          julia --project=test --color=yes -e '
+            import Pkg
+            Pkg.develop(Pkg.PackageSpec(path=pwd()))
+            Pkg.instantiate()
+            Pkg.status(; mode = Pkg.PKGMODE_MANIFEST)
+            include("test/runtests.jl")
+          '
diff --git a/test/cmesh/test_readmshfile.jl b/test/cmesh/test_readmshfile.jl
index 94d6729..a649757 100644
--- a/test/cmesh/test_readmshfile.jl
+++ b/test/cmesh/test_readmshfile.jl
@@ -70,7 +70,7 @@ end
 
 @testset "readmshfile" begin
     @testset "test_msh_file_vers2_ascii" begin
-        fileprefix = "cmesh/testfiles/test_msh_file_vers2_ascii"
+        fileprefix = joinpath(@__DIR__, "testfiles", "test_msh_file_vers2_ascii")
         filename = fileprefix * ".msh"
 
         @assert isfile(filename) "File not found: "*filename
@@ -84,7 +84,7 @@ end
     end
 
     @testset "test_msh_file_vers4_ascii" begin
-        fileprefix = "cmesh/testfiles/test_msh_file_vers4_ascii"
+        fileprefix = joinpath(@__DIR__, "testfiles", "test_msh_file_vers4_ascii")
         filename = fileprefix * ".msh"
 
         @assert isfile(filename) "File not found: "*filename
@@ -97,7 +97,7 @@ end
     end
 
     @testset "test_msh_file_vers2_bin" begin
-        fileprefix = "cmesh/testfiles/test_msh_file_vers2_bin"
+        fileprefix = joinpath(@__DIR__, "testfiles", "test_msh_file_vers2_bin")
         filename = fileprefix * ".msh"
 
         @assert isfile(filename) "File not found: "*filename
@@ -112,7 +112,7 @@ end
     end
 
     @testset "test_msh_file_vers4_bin" begin
-        fileprefix = "cmesh/testfiles/test_msh_file_vers4_bin"
+        fileprefix = joinpath(@__DIR__, "testfiles", "test_msh_file_vers4_bin")
         filename = fileprefix * ".msh"
 
         @assert isfile(filename) "File not found: "*filename
diff --git a/test/runtests.jl b/test/runtests.jl
index 9ca3763..389f6f3 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -17,7 +17,7 @@ import MPIPreferences
 
         @info "Starting parallel tests"
 
-        run(`$(mpiexec()) -n 2 $(Base.julia_cmd()) --threads=1 --check-bounds=yes --project=$(dirname(@__DIR__)) $(abspath("test_all.jl"))`)
+        run(`$(mpiexec()) -n 2 $(Base.julia_cmd()) --threads=1 --check-bounds=yes --project=$(dirname(@__DIR__)) $(joinpath(@__DIR__, "test_all.jl"))`)
 
         @info "Finished parallel tests"
     end