The local weight-monodromy conjecture for the mixed central functor

Let $V$ be a representation of the Langlands dual group, and let $\scrZ^{\mathrm{mix}}(V)$ be the associated mixed perverse sheaf obtained from the mixed central functor. Its monodromy filtration is the filtration induced by the nilpotent monodromy operator, and its weight filtration is the filtration by mixed perverse sheaves with pure weight-$i$ graded pieces. The sheaf is monodromy-pure of weight $0$. The monodromy and weight filtrations on $\scrZ^{\mathrm{mix}}(V)$ should coincide. This is the local weight-monodromy conjecture in the setting of the mixed central functor; the supplied text does not state a resolution.