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We consider the set of smooth zero degree mapsψ  :      2   →     2{\psi\colon\mathbb{S}^{2}\to\mathbb{S}^{2}}    which have the following properties: (i) There is a unique minimizing harmonic mapu  :      3   →     2{u\colon\mathbb{B}^{3}\to\mathbb{S}^{2}}    which agrees with ψ on the boundary of the unit ball. (ii) The map  u  has at least  N  singular points in  3    {\mathbb{B}^{3}}   . (iii) The Lavrentiev gap phenomenon holds for ψ, i.e., the infimum of the Dirichlet energiesE  ⁢   (  w  ){E(w)}    of all smooth extensionsw  :      3   →     2{w\colon\mathbb{B}^{3}\to\mathbb{S}^{2}}    of ψ is strictly larger than the Dirichlet energy∫     3|  ∇  u   |  2{\int_{\mathbb{B}^{3}}\lvert\nabla u|^{2}}    of the (irregular) minimizer  u . For each positive integer  N , we prove that this set is dense in the set   {\mathcal{S}}    of all smooth zero degree mapsϕ  :      2   →     2{\phi\colon\mathbb{S}^{2}\to\mathbb{S}^{2}}    endowed with theW   1  ,  p{W^{1,p}}   -topology, where 1≤   \leq    p  ¡ 2. This result is sharp since it fails in theW   1  ,  2{W^{1,2}}   -topology of   {\mathcal{S}}   .

The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data