Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}) and conductor c = max(N \ S) + 1. Let P be the set of primitive elements of S, and let L be the set of elements of S which are smaller than c. Wilf's conjecture (1978) states that the inequality |P||L| ≥ c always hold. The conjecture has been shown to hold in case |P| ≥ m/2 by Sammartano in 2012, and subsequently in case |P| ≥ m/3 by the author in 2020. The main result in this paper is that Wilf's conjecture holds in case |P| ≥ m/4 with c ∈ mN. 0 Caution This document is the starting point of a full paper to be gradually completed in successive versions within the next few weeks of Fall 2023.