• We consider spanning trees and their co-trees in 3-connected planar graphs. • We give a spanning tree such that the tree and its co-tree have maximum degree 4. • Additionally, we provide an upper bound on the number of vertices of degree 4. • This is the best known result toward a solution of a conjecture of Gr̎unbaum of 1970. Given a spanning tree T of a planar graph G , the co-tree of T is the spanning tree of the dual graph G * with edge set ( E ( G ) − E ( T ) ) * . Grünbaum conjectured in 1970 that every planar 3-connected graph G contains a spanning tree T such that both T and its co-tree have maximum degree at most 3. While Grünbaum’s conjecture remains open, Schmidt and the author recently improved the upper bound on the maximum degree from 5 (Biedl 2014) to 4. In this paper, we modify this approach taking a further step towards Grünbaum’s conjecture. We again obtain a spanning tree T such that both T and its co-tree have maximum degree at most 4 and, additionally, an upper bound on the number of vertices of degree 4 of T and its co-tree.