The line hypergraphs of generalized Petersen graphs GP(5k+11,2) form infinitely many counterexamples to Lovász's conjecture for r=3, and several quartic graphs give counterexamples for r=4.
Ryser’s Conjecture for t-intersecting hypergraphs
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Infinitely many counterexamples to a conjecture of Lov\'asz
The line hypergraphs of generalized Petersen graphs GP(5k+11,2) form infinitely many counterexamples to Lovász's conjecture for r=3, and several quartic graphs give counterexamples for r=4.