diff --git a/Project.toml b/Project.toml
index d2b37d0d5..e5f091eec 100644
--- a/Project.toml
+++ b/Project.toml
@@ -25,11 +25,11 @@ DocStringExtensions = "0.8"
 GenericLinearAlgebra = "0.2"
 IterativeSolvers = "0.8, 0.9"
 LinearMaps = "2.7, 3"
-MathOptInterface = "0.10"
+MathOptInterface = "0.10.6"
 PolynomialRoots = "1"
 Requires = "1"
 SpecialFunctions = "1.4, 2"
-julia = "1.5"
+julia = "1.6"
 
 [extras]
 DynamicPolynomials = "7c1d4256-1411-5781-91ec-d7bc3513ac07"
diff --git a/src/MathOptInterface/wrapper.jl b/src/MathOptInterface/wrapper.jl
index e5bfcea23..4117d0e26 100644
--- a/src/MathOptInterface/wrapper.jl
+++ b/src/MathOptInterface/wrapper.jl
@@ -110,7 +110,7 @@ function MOI.copy_to(
     # objective
     opt.obj_sense = MOI.MIN_SENSE
     model_c = zeros(T, n)
-    obj_offset = 0.0
+    obj_offset = zero(T)
     for attr in MOI.get(src, MOI.ListOfModelAttributesSet())
         if attr == MOI.Name()
             continue
diff --git a/test/runmoitests.jl b/test/runmoitests.jl
index 1660749e0..8c1f5c58d 100644
--- a/test/runmoitests.jl
+++ b/test/runmoitests.jl
@@ -25,14 +25,19 @@ include(joinpath(@__DIR__, "moicones.jl"))
     end
 end
 
-@testset "MOI.Test tests" begin
-    println("\nstarting MOI.Test tests")
-    # TODO test other real types
-    T = Float64
+# real types, tolerances, and tests to include for MOI.Test tests
+test_T = [
+    (Float64, 2 * sqrt(sqrt(eps())), 4, String[]),
+    # TODO add test_linear after MOI 0.10.7 is tagged:
+    (BigFloat, 2 * eps(BigFloat)^0.2, 1, String["test_conic"]),
+]
+
+@testset "MOI.Test tests: $T" for (T, tol_test, tol_relax, includes) in test_T
+    println("\nstarting MOI.Test tests: $T")
     model = MOI.Bridges.full_bridge_optimizer(
         MOI.Utilities.CachingOptimizer(
             MOI.Utilities.UniversalFallback(MOI.Utilities.Model{T}()),
-            Hypatia.Optimizer{T}(; default_tol_relax = 4),
+            Hypatia.Optimizer{T}(; default_tol_relax = tol_relax),
         ),
         T,
     )
@@ -41,8 +46,8 @@ end
         model,
         MOI.Test.Config(
             T,
-            atol = 2 * sqrt(sqrt(eps(T))),
-            rtol = 2 * sqrt(sqrt(eps(T))),
+            atol = tol_test,
+            rtol = tol_test,
             exclude = Any[
                 MOI.ConstraintBasisStatus,
                 MOI.VariableBasisStatus,
@@ -50,10 +55,9 @@ end
                 MOI.SolverVersion,
             ],
         ),
-        # include = String["test_conic"],
+        include = includes,
         exclude = String[
-            # TODO(odow): unexpected failure. But this is probably in the bridge
-            # layer, not Hypatia.
+            # TODO(odow): unexpected failure, probably in the bridge layer
             "test_model_UpperBoundAlreadySet",
             "test_model_LowerBoundAlreadySet",
         ],