Poincare duality for algebraic de Rham cohomology with integrable connection coefficients

Question

I am reading "The Gauss-Manin Connection and Tannaka Duality" (here is the link to the paper). I am specifically interested in the proof of Proposition 2.2. In this proof, the authors use the Poincaré duality for algebraic de Rham cohomology of a smooth projective curve $X$ over a field $K$ of characteristic $0$ with integrable connection coefficient $(\mathcal V,\bigtriangledown)$. You can see this line lies between equations (2.12) and (2.13) in the paper:

$H^2_{dR}(X,(\mathcal V, \bigtriangledown)) $ is a Poincaré dual to $H^0_{dR}(X,(\mathcal V,\bigtriangledown)^\vee)$, where $(\mathcal V, \bigtriangledown)^\vee$ is the dual connection.

However, I can not find any references for proof of this claim. In the Stacks project (here), this duality is only proven for trivial connections $(\Omega ^ p_{X/k},d)$. So I wonder whether any of you have a reference for this more general duality.

Thanks a lot!

Answer 1

It appears in the book "De Rham Cohomology of Differential Modules on Algebraic Varieties" of Yves André, Francesco Baldassarri, and Maurizio Cailotto, Theorem D.2.17.