diff --git a/Project.toml b/Project.toml
index 830c0f57a..566025b33 100644
--- a/Project.toml
+++ b/Project.toml
@@ -73,8 +73,8 @@ LogDensityProblemsAD = "1.7.0"
 MCMCChains = "5, 6"
 NamedArrays = "0.9, 0.10"
 Optim = "1"
-Optimization = "3"
-OptimizationOptimJL = "0.1, 0.2, 0.3"
+Optimization = "3, 4"
+OptimizationOptimJL = "0.1, 0.2, 0.3, 0.4"
 OrderedCollections = "1"
 Printf = "1"
 Random = "1"
diff --git a/test/Project.toml b/test/Project.toml
index d5d83726f..e68947a68 100644
--- a/test/Project.toml
+++ b/test/Project.toml
@@ -56,10 +56,10 @@ MCMCChains = "5, 6"
 Mooncake = "0.4.19"
 NamedArrays = "0.9.4, 0.10"
 Optim = "1"
-Optimization = "3"
-OptimizationBBO = "0.1, 0.2, 0.3"
-OptimizationNLopt = "0.1, 0.2"
-OptimizationOptimJL = "0.1, 0.2, 0.3"
+Optimization = "3, 4"
+OptimizationBBO = "0.1, 0.2, 0.3, 0.4"
+OptimizationNLopt = "0.1, 0.2, 0.3"
+OptimizationOptimJL = "0.1, 0.2, 0.3, 0.4"
 PDMats = "0.10, 0.11"
 Pkg = "1"
 Random = "1"
diff --git a/test/optimisation/Optimisation.jl b/test/optimisation/Optimisation.jl
index d8afd83db..a5d4f994a 100644
--- a/test/optimisation/Optimisation.jl
+++ b/test/optimisation/Optimisation.jl
@@ -624,14 +624,7 @@ using Turing
         m = DynamicPPL.contextualize(
             gdemo_default, ADUtils.ADTypeCheckContext(adbackend, gdemo_default.context)
         )
-        if adbackend isa AutoMooncake
-            # Optimization.jl does not support Mooncake as an AD backend, see
-            # https://docs.sciml.ai/Optimization/stable/API/ad/#ad
-            # If it ever does, then we should just run them to make sure they don't error
-            err_msg = "The passed automatic differentiation backend choice is not available"
-            @test_throws err_msg maximum_likelihood(m; adtype=adbackend)
-            @test_throws err_msg maximum_a_posteriori(m; adtype=adbackend)
-        elseif adbackend isa AutoForwardDiff
+        if adbackend isa AutoForwardDiff
             # TODO: Figure out why this is happening.
             # https://github.com/TuringLang/Turing.jl/issues/2369
             @test_throws DivideError maximum_likelihood(m; adtype=adbackend)