diff --git a/packaging/conda/build.sh b/packaging/conda/build.sh
index 85a4d1174..80357b5bb 100644
--- a/packaging/conda/build.sh
+++ b/packaging/conda/build.sh
@@ -4,20 +4,31 @@
 set -euo pipefail
 
 # Choose the build preset for macOS/Linux
-if [[ "$(uname)" == "Darwin" ]]
+if [[ "$(uname)" == "Linux" ]]
 then
-  CMAKE_PRESET="macos-clang-release"
-else
   CMAKE_PRESET="linux-gcc-release"
+  cmake --fresh --preset $CMAKE_PRESET -DBUILD_JARS=OFF -DTESTING=OFF -DMPI_SUFFIX=_mpich
+else
+  CMAKE_PRESET="macos-clang-release"
+  cmake --fresh --preset $CMAKE_PRESET -DBUILD_JARS=OFF -DTESTING=OFF
 fi
 
 # Build MODL and MODL_Coclustering
-cmake --fresh --preset $CMAKE_PRESET -DBUILD_JARS=OFF -DTESTING=OFF
 cmake --build --preset $CMAKE_PRESET --parallel --target MODL MODL_Coclustering
 
-# Copy the MODL binaries to the Conda PREFIX path
-cp "./build/$CMAKE_PRESET/bin/MODL" "$PREFIX/bin"
-cp "./build/$CMAKE_PRESET/bin/MODL_Coclustering" "$PREFIX/bin"
+# Move the MODL binaries to the Conda PREFIX path
+mv ./build/$CMAKE_PRESET/bin/MODL* "$PREFIX/bin"
+
+# Rename MODL in Linux since it has a prefix depending of the MPI version
+if [[ "$(uname)" == "Linux" ]]
+then
+  if [[ -f "$PREFIX/bin/MODL_mpich" ]]
+  then
+    mv "$PREFIX/bin/MODL_mpich" "$PREFIX/bin/MODL"
+  else
+    mv "$PREFIX/bin/MODL_openmpi" "$PREFIX/bin/MODL"
+  fi
+fi
 
 # Custom rpath relocation and signing executables for macOS in arm64
 #
@@ -30,8 +41,8 @@ cp "./build/$CMAKE_PRESET/bin/MODL_Coclustering" "$PREFIX/bin"
 # application pressing on "Allow" works).
 #
 # However, in the default settings, `conda build` relocalizes the executable by changing rpath of
-# the library paths at $PREFIX by relative ones and in doing so it nullifies any signature. So we
-# do ourselves this procedure first and then sign the binary.
+# the library paths at $PREFIX by relative ones and in doing so it nullifies any signature. So we do
+# ourselves this procedure first and then sign the binary.
 #
 # Note that in meta.yaml for osx-arm64 we have custom build.binary_relocation and
 # build.detect_binary_files_with_prefix option
diff --git a/packaging/conda/conda_build_config.yaml b/packaging/conda/conda_build_config.yaml
index 03d8c9e6a..185e09f7c 100644
--- a/packaging/conda/conda_build_config.yaml
+++ b/packaging/conda/conda_build_config.yaml
@@ -1,4 +1,30 @@
 ---
+# Note on MPI version pinning:
+# We pin the MPI versions so the build does not suddently stop working
+# Linux:
+# - mpich=4.0.3 because of bugs of the 3.* series
+# - requires conda-forge
+# OSX:
+# - mpich=3.4.3 because 4.* is still unstable
+# - requires conda-forge
+# Windows:
+# - msmpi=10.1.1 is the latest as of this commit
+mpi:
+  - mpich=4.0.3  # [linux]
+  - mpich=3.4.3  # [osx]
+  - msmpi=10.1.1  # [win]
+c_compiler:
+  - gcc  # [linux]
+  - clang  # [osx]
+c_compiler_version:
+  - 12  # [linux]
+  - 16  # [osx]
+cxx_compiler:
+  - gxx  # [linux]
+  - clangxx  # [osx]
+cxx_compiler_version:
+  - 12  # [linux]
+  - 16  # [osx]
 # We need MacOS SDK 10.10 to be able to build on macOS Intel
 # Download: https://github.com/phracker/MacOSX-SDKs/releases/download/11.3/MacOSX10.10.sdk.tar.xz
 # Decompress then to /opt: tar -zxvf MacOSX10.10.sdk.tar.xz -C /opt
diff --git a/packaging/conda/meta.yaml b/packaging/conda/meta.yaml
index 662716986..0cc60d079 100644
--- a/packaging/conda/meta.yaml
+++ b/packaging/conda/meta.yaml
@@ -26,35 +26,15 @@ build:
   detect_binary_files_with_prefix: false # [osx]
   {% endif %}
 
-# Note on version pinning:
-# OSX:
-# - mpich=3.4.3 because 4.* is still unstable
-# - requires conda-forge
-# Linux:
-# - mpich=4.0.3 because of bugs of the 3.* series
-# - requires conda-forge
 requirements:
   build:
-    - mpich 4.0.3        # [linux]
-    - mpich-mpicc  4.0.3 # [linux]
-    - mpich-mpicxx 4.0.3 # [linux]
-    - mpich 3.4.3        # [osx]
-    - mpich-mpicc  3.4.3 # [osx]
-    - mpich-mpicxx 3.4.3 # [osx]
-    - msmpi              # [win]
     - cmake
     - ninja
     - {{ compiler('cxx') }}
   host:
-    - mpich 4.0.3        # [linux]
-    - mpich-mpicxx 4.0.3 # [linux]
-    - mpich 3.4.3        # [osx]
-    - mpich-mpicxx 3.4.3 # [osx]
-    - msmpi              # [win]
+    - {{ mpi }}
   run:
-    - mpich 4.0.3   # [linux]
-    - mpich 3.4.3   # [osx]
-    - msmpi         # [win]
+    - {{ mpi }}
 
 outputs:
   - name: khiops-core