  We settle the Hadwiger-Boltyanski Illumination Conjecture for all
1-unconditional convex bodies in ${\mathbb R}^3$ and in ${\mathbb R}^4$.
Moreover, we settle the conjecture for those higher-dimensional 1-unconditional
convex bodies which have at least one coordinate hyperplane projection equal to
the corresponding projection of the circumscribing rectangular box. Finally, we
confirm the conjectured equality cases of the Illumination Conjecture within
the subclass of 1-unconditional bodies which, just like the cube $[-1,1]^n$,
have no extreme points on coordinate subspaces.
  Our methods are combinatorial, and the illuminating sets that we use consist
primarily of small perturbations of the standard basis vectors. In particular,
we build on ideas and constructions from [Sun-Vritsiou, "On the illumination of
1-symmetric convex bodies", preprint available at arXiv:2407.10314], and mainly
on the notion of 'deep illumination' introduced there.
