Compare the Theorem 1.1 isomorphism with Schefers’ perverse-sheaf isomorphism

Determine whether the isomorphism between the bi-filtered D_{X×A^t}-module Ø_g(T'(M)[-r], F, W) and the microlocalization (u_z(M), F, W) established in Theorem 1.1 agrees, via the Riemann–Hilbert correspondence, with the isomorphism between vanishing cycles and microlocalization constructed by K. Schefers for perverse sheaves; in particular, prove that these isomorphisms coincide so that the Theorem 1.1 isomorphism upgrades to a canonical isomorphism of Q-mixed Hodge modules for all mixed Hodge modules M on X.

Background

Theorem 1.1 in this paper provides an isomorphism at the level of bi-filtered D-modules underlying mixed Hodge modules, identifying vanishing cycles along g = ∑ y_i f_i of T'(M)[-r] with the microlocalization u_z(M) obtained via Verdier specialization followed by Fourier–Laplace transform.

K. Schefers [Sch22] proved an equivalence between vanishing cycles and microlocalization for perverse sheaves. The authors suspect the two isomorphisms match, which would imply compatibility of the constructions and allow an enhancement from a D-module-level statement to an isomorphism in the category of Q-mixed Hodge modules.