diff --git a/.ci/scripts/test_old_deps.sh b/.ci/scripts/test_old_deps.sh
index 769ca4517edf..7d0625fa86f7 100755
--- a/.ci/scripts/test_old_deps.sh
+++ b/.ci/scripts/test_old_deps.sh
@@ -27,9 +27,10 @@ export VIRTUALENV_NO_DOWNLOAD=1
 
 # Patch the project definitions in-place:
 # - Replace all lower and tilde bounds with exact bounds
-# - Make the pyopenssl 17.0, which is the oldest version that works with
-#   a `cryptography` compiled against OpenSSL 1.1.
+# - Replace all caret bounds---but not the one that defines the supported Python version!
 # - Delete all lines referring to psycopg2 --- so no testing of postgres support.
+# - Use pyopenssl 17.0, which is the oldest version that works with
+#   a `cryptography` compiled against OpenSSL 1.1.
 # - Omit systemd: we're not logging to journal here.
 
 # TODO: also replace caret bounds, see https://python-poetry.org/docs/dependency-specification/#version-constraints
@@ -40,6 +41,7 @@ export VIRTUALENV_NO_DOWNLOAD=1
 
 sed -i \
    -e "s/[~>]=/==/g" \
+   -e '/^python = "^/!s/\^/==/g' \
    -e "/psycopg2/d" \
    -e 's/pyOpenSSL = "==16.0.0"/pyOpenSSL = "==17.0.0"/' \
    -e '/systemd/d' \
diff --git a/changelog.d/13082.misc b/changelog.d/13082.misc
new file mode 100644
index 000000000000..1aa386dbf733
--- /dev/null
+++ b/changelog.d/13082.misc
@@ -0,0 +1 @@
+Pin dependencies maintained by matrix.org to [semantic version](https://semver.org/) bounds.
diff --git a/poetry.lock b/poetry.lock
index 849e8a7a99e4..49fbaab57782 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -1563,7 +1563,7 @@ url_preview = ["lxml"]
 [metadata]
 lock-version = "1.1"
 python-versions = "^3.7.1"
-content-hash = "73882e279e0379482f2fc7414cb71addfd408ca48ad508ff8a02b0cb544762af"
+content-hash = "e96625923122e29b6ea5964379828e321b6cede2b020fc32c6f86c09d86d1ae8"
 
 [metadata.files]
 attrs = [
diff --git a/pyproject.toml b/pyproject.toml
index 44aa775c33b7..3a56c42c0b1f 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -110,9 +110,9 @@ jsonschema = ">=3.0.0"
 frozendict = ">=1,!=2.1.2"
 # We require 2.1.0 or higher for type hints. Previous guard was >= 1.1.0
 unpaddedbase64 = ">=2.1.0"
-canonicaljson = ">=1.4.0"
+canonicaljson = "^1.4.0"
 # we use the type definitions added in signedjson 1.1.
-signedjson = ">=1.1.0"
+signedjson = "^1.1.0"
 # validating SSL certs for IP addresses requires service_identity 18.1.
 service-identity = ">=18.1.0"
 # Twisted 18.9 introduces some logger improvements that the structured
@@ -150,7 +150,7 @@ typing-extensions = ">=3.10.0.1"
 cryptography = ">=3.4.7"
 # ijson 3.1.4 fixes a bug with "." in property names
 ijson = ">=3.1.4"
-matrix-common = "~=1.2.1"
+matrix-common = "^1.2.1"
 # We need packaging.requirements.Requirement, added in 16.1.
 packaging = ">=16.1"
 # At the time of writing, we only use functions from the version `importlib.metadata`