$\mathfrak{sl}(2)$-weight system does not extend to a graph 4-invariant

A long-standing question by S. Lando asks whether the $\mathfrak{sl}(2)$-weight system extends to a unique 4-invariant of graphs. We show that, in full generality, the answer to this question is negative. However, for certain specializations of the weight system, extensions do exist. Explicit formulae for computing two such specializations of the weight system are already known. We construct recurrence relations for one additional such extension and discuss the last remaining specialization, which conjecturally admits an extension. We also study the polynomial coefficients of the $\mathfrak{sl}(2)$-weight system and resolve the questions concerning their extension.