Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in r-partite r-uniform hypergraphs, Lovász formulated a stronger conjecture. It states that one can always reduce the matching number by removing r-1 vertices. This conjecture was very recently disproven for r=3 by Clow, Haxell, and Mohar using the line graph of a 3-regular graph of order 102. Building on this, we describe a simple infinite family of counterexamples based on generalized Petersen graphs for the case r=3 and give specific counterexamples for r=4.
Abiad et al. (Thu,) studied this question.