Higher-order $p$-form asymptotic symmetries in $D = p + 2$

We investigate higher-order asymptotic symmetries for a $p$-form gauge field in $(p + 2)$-dimensional Minkowski spacetime, where Hodge duality with a scalar holds. Employing symplectic renormalization, we identify $N + 1$ independent asymptotic charges, with each charge being parametrised by an arbitrary function of the angular variables. By means of the Hodge decomposition, these charges share the same formal structure independently from p and are manifestly dual to a scalar charge. We work in Lorenz gauge, therefore the gauge parameters require a radial expansion involving logarithmic (subleading) terms to ensure nontrivial angular dependence at leading order. At the same time we assume a power expansion for the field strength, allowing logarithms in the gauge field expansions within pure gauge sectors.