Tunnel number, connected sum and meridional essential surfaces

For given orientable closed 3-manifolds M1, M2,…, Mn and for given knots K1, K2,…, Kn in M1, M2,…, Mn, respectively, we show that if none of those 3-manifolds have lens space summands and none of those knot exteriors contain meridional essential surfaces, then the tunnel numbers of those knots do not go down under connected sum, i.e. t(K1#K2#…#Kn)⩾t(K1)+t(K2)+…+t(Kn).