diff --git a/src/FastAlmostBandedMatrices.jl b/src/FastAlmostBandedMatrices.jl
index 6d0ca48..20119d8 100644
--- a/src/FastAlmostBandedMatrices.jl
+++ b/src/FastAlmostBandedMatrices.jl
@@ -2,8 +2,15 @@ module FastAlmostBandedMatrices
 
 import PrecompileTools: @setup_workload, @compile_workload
 
-using ArrayInterface, ArrayLayouts, ConcreteStructs, LazyArrays, LinearAlgebra,
-    MatrixFactorizations
+import ArrayInterface
+import ArrayLayouts
+import ArrayLayouts: LayoutMatrix, LayoutVector, Ldiv, Lmul, TriangularLayout
+import ConcreteStructs: @concrete
+import LazyArrays: LazyArray, Mul
+import LinearAlgebra
+import LinearAlgebra: LowerTriangular, NoPivot, UnitLowerTriangular, UnitUpperTriangular,
+    UpperTriangular, diagind, lmul!, lu, qr, rank, triu!
+import MatrixFactorizations
 
 # The BandedMatrices.jl surface that FastAlmostBandedMatrices reexports (see the second
 # `export` below), so that `using FastAlmostBandedMatrices` on its own is enough to build
@@ -640,22 +647,26 @@ end
 
 _almostbanded_widerect_ldiv!(::QR{T}, B) where {T} = error("Not implemented")
 
-const UpperLayoutMatrix{T} = UpperTriangular{T, <:LayoutMatrix{T}}
-
-for Typ in
-    (:StridedVector, :StridedMatrix, :AbstractVecOrMat, :UpperLayoutMatrix, :LayoutMatrix)
-    @eval function ldiv!(A::QR{T, <:AlmostBandedMatrix}, B::$Typ{T}) where {T}
-        m, n = size(A)
-        return if m == n
-            _almostbanded_square_ldiv!(A, B)
-        elseif n > m
-            _almostbanded_widerect_ldiv!(A, B)
-        else
-            _almostbanded_longrect_ldiv!(A, B)
-        end
+function _almostbanded_ldiv!(A::QR, B)
+    m, n = size(A)
+    return if m == n
+        _almostbanded_square_ldiv!(A, B)
+    elseif n > m
+        _almostbanded_widerect_ldiv!(A, B)
+    else
+        _almostbanded_longrect_ldiv!(A, B)
     end
 end
 
+ldiv!(A::QR{T, <:AlmostBandedMatrix}, B::StridedVector{T}) where {T} =
+    _almostbanded_ldiv!(A, B)
+ldiv!(A::QR{T, <:AlmostBandedMatrix}, B::StridedMatrix{T}) where {T} =
+    _almostbanded_ldiv!(A, B)
+ldiv!(A::QR{T, <:AlmostBandedMatrix}, B::LayoutVector{T}) where {T} =
+    _almostbanded_ldiv!(A, B)
+ldiv!(A::QR{T, <:AlmostBandedMatrix}, B::LayoutMatrix{T}) where {T} =
+    _almostbanded_ldiv!(A, B)
+
 # needed for adaptive QR
 function Base.materialize!(M::Lmul{<:QRPackedQLayout{<:AlmostBandedLayout}})
     return lmul!(QRPackedQ(bandpart(M.A.factors), M.A.τ), M.B)
@@ -670,6 +681,12 @@ triangularlayout(::Type{Tri}, ::ML) where {Tri, ML <: AlmostBandedLayout} = Tri{
 @inline function __arguments(x::LazyArray, ::AlmostBandedMatrix, ::Val)
     return LazyArrays.arguments(x)
 end
+@inline function __arguments(x::LazyArray, ::AlmostBandedMatrix, ::Val{false})
+    return LazyArrays.arguments(x)
+end
+@inline function __arguments(x::LazyArray, ::AlmostBandedMatrix, ::Val{true})
+    return LazyArrays.arguments(x)
+end
 @inline __arguments(x::Mul, ::AlmostBandedMatrix, ::Val) = (x.A, x.B)
 @inline __arguments(x::AbstractArray, ::AlmostBandedMatrix, ::Val{false}) = (nothing, x)
 @inline function __arguments(x::AbstractArray, A::AlmostBandedMatrix, ::Val{true})
diff --git a/test/core_tests.jl b/test/core_tests.jl
index 14729fd..8fc8035 100644
--- a/test/core_tests.jl
+++ b/test/core_tests.jl
@@ -173,6 +173,8 @@ end
     @test A \ b ≈ Matrix(A) \ b
     @test all(A \ b .=== F \ b)
     @test all(A \ b .=== F.R \ (F.Q' * b))
+    @test ldiv!(F, copy(b)) ≈ Matrix(A) \ b
+    @test ldiv!(F, reshape(copy(b), :, 1)) ≈ reshape(Matrix(A) \ b, :, 1)
     Q̃ = QRPackedQ(F.factors, F.τ)
     @test Matrix(Q̃) ≈ Matrix(F.Q)
     @test lmul!(Q̃, copy(b)) ≈ lmul!(F.Q, copy(b)) ≈ Matrix(F.Q) * b
diff --git a/test/qa/qa.jl b/test/qa/qa.jl
index a77dba6..6a9e2ce 100644
--- a/test/qa/qa.jl
+++ b/test/qa/qa.jl
@@ -15,10 +15,6 @@ run_qa(
     # `Band` and `BandError` are reexported but carry no docstring upstream in
     # BandedMatrices.jl, so the public-API docstring check has nothing to find for them.
     api_docs_kwargs = (; ignore = (:Band, :BandError)),
-    # 19 method ambiguities, all in FastAlmostBandedMatrices' own ldiv!/__arguments
-    # against ArrayLayouts/LinearAlgebra Triangular/Factorization methods.
-    # https://github.com/SciML/FastAlmostBandedMatrices.jl/issues/71
-    aqua_broken = (:ambiguities,),
     ei_kwargs = (;
         # Non-public names this package legitimately extends/uses from upstream:
         # ArrayLayouts MatLdivVec/sublayout/triangulardata/triangularlayout/_qr/_qr!/
@@ -42,9 +38,4 @@ run_qa(
             ),
         ),
     ),
-    # 31 names implicitly imported via the package's `using ArrayInterface, ArrayLayouts,
-    # ConcreteStructs, LazyArrays, LinearAlgebra, ...`; explicit-import conversion tracked
-    # separately.
-    # https://github.com/SciML/FastAlmostBandedMatrices.jl/issues/71
-    ei_broken = (:no_implicit_imports,),
 )