diff --git a/benchmark/2_bench_state_estim.jl b/benchmark/2_bench_state_estim.jl
index 982aeeaa2..c53c0e444 100644
--- a/benchmark/2_bench_state_estim.jl
+++ b/benchmark/2_bench_state_estim.jl
@@ -146,6 +146,7 @@ UNIT_ESTIM["ExtendedKalmanFilter"]["evaloutput"]["NonLinModel"] =
 
 mhe_lin_curr    = MovingHorizonEstimator(linmodel, He=10, direct=true)
 mhe_lin_pred    = MovingHorizonEstimator(linmodel, He=10, direct=false)
+mhe_lin_skf     = MovingHorizonEstimator(linmodel, He=10, covestim=SteadyKalmanFilter(linmodel))
 mhe_nonlin_curr = MovingHorizonEstimator(nonlinmodel, He=10, direct=true)
 mhe_nonlin_pred = MovingHorizonEstimator(nonlinmodel, He=10, direct=false)
 
@@ -161,6 +162,12 @@ UNIT_ESTIM["MovingHorizonEstimator"]["updatestate!"]["LinModel"]["Current form"]
         setup=preparestate!($mhe_lin_curr, $y, $d),
         samples=samples, evals=evals, seconds=seconds,
     )
+UNIT_ESTIM["MovingHorizonEstimator"]["updatestate!"]["LinModel"]["Constant arr. cov."] = 
+    @benchmarkable(
+        updatestate!($mhe_lin_skf, $u, $y, $d),
+        setup=preparestate!($mhe_lin_skf, $y, $d),
+        samples=samples, evals=evals, seconds=seconds,
+    )
 UNIT_ESTIM["MovingHorizonEstimator"]["preparestate!"]["LinModel"]["Prediction form"] =
     @benchmarkable(
         preparestate!($mhe_lin_pred, $y, $d),
diff --git a/check_syntax.jl b/check_syntax.jl
new file mode 100644
index 000000000..fcb2c74ce
--- /dev/null
+++ b/check_syntax.jl
@@ -0,0 +1,2 @@
+import JuliaFormatter
+include("src/estimator/mhe/execute.jl")
diff --git a/src/estimator/mhe/execute.jl b/src/estimator/mhe/execute.jl
index c5f8a515d..670fdb953 100644
--- a/src/estimator/mhe/execute.jl
+++ b/src/estimator/mhe/execute.jl
@@ -709,7 +709,7 @@ function correct_cov!(estim::MovingHorizonEstimator)
         correct_estimate!(estim.covestim, y0marr, d0arr)
         all(isfinite, estim.covestim.cov.P̂) || error("Arrival covariance P̄ is not finite")
         estim.P̂arr_old .= estim.covestim.cov.P̂
-        invert_cov!(estim, estim.P̂arr_old)
+        update_arrival_cov!(estim)
     catch err
         if err isa PosDefException
             @error("Arrival covariance P̄ is not positive definite: keeping the old one")
@@ -736,7 +736,7 @@ function update_cov!(estim::MovingHorizonEstimator)
         update_estimate!(estim.covestim, y0marr, d0arr, u0arr)
         all(isfinite, estim.covestim.cov.P̂) || error("Arrival covariance P̄ is not finite")
         estim.P̂arr_old .= estim.covestim.cov.P̂
-        invert_cov!(estim, estim.P̂arr_old)
+        update_arrival_cov!(estim)
     catch err
         if err isa PosDefException
             @error("Arrival covariance P̄ is not positive definite: keeping the old one")
@@ -764,6 +764,19 @@ function invert_cov!(estim::MovingHorizonEstimator, P̄)
     return nothing
 end
 
+"Update the arrival covariance matrix at the next time step based on the covariance estimator type."
+function update_arrival_cov!(estim::MovingHorizonEstimator)
+    _update_arrival_cov!(estim, estim.covestim)
+end
+
+function _update_arrival_cov!(estim::MovingHorizonEstimator, ::StateEstimator)
+    invert_cov!(estim, estim.P̂arr_old)
+end
+
+function _update_arrival_cov!(estim::MovingHorizonEstimator, ::SteadyKalmanFilter)
+    return nothing
+end
+
 """
     obj_nonlinprog(estim::MovingHorizonEstimator, ::LinModel, _ , _ , _ , Z̃)