Sparse backslash

I get a StackOverflowError if I try to differentiate some code with a sparse matrix dependent on the variable and I use backslash with it. MWE:

using SparseArrays, ForwardDiff, LinearAlgebra
A = sparse([1.0 1.0 ; 0.0 1.0])
f(x) = sparse(A + x * x') \ ones(2)
x = ones(2)
f(x)
ForwardDiff.jacobian(f, x)

Compared to the "full" version, which works:

f_full(x) = (A + x * x') \ ones(2)
x = ones(2)
f_full(x)
ForwardDiff.jacobian(f_full, x)

I feel like this could be solved by adding a rule for underlying machinery of dual numbers for backslash, something akin to

function Base.:\(M::Dual{T}, x)
    MRf = factorize(realpart.(M))
    sol = MRf \ x
    sol -= Dual(0.0, 1.0) * (MRf \ (dualpart.(M) * sol))
    return sol
end

(That's what I do when I use DualNumbers.jl.) This should work and allow to only use LinearAlgebra with real-valued types because

But I am not sure how to suggest changes directly in ForwardDiff.jl. Does that make sense?

Start with the provided MWE at the ForwardDiff.jacobian entry point and compare the sparse \ path with the working full-matrix path. Reproduce the StackOverflowError, then verify that differentiation of the sparse solve completes and returns a Jacobian without changing the full-matrix behavior.