Does SOC has no effects on orbitals with different $l$?

SOC term is:
$$H_{mathrm{SOC}}=alpha vec{L} cdot vec{S}=frac{alpha}{2}left(L^{+} sigma^{+}+L^{-} sigma^{-}+L^{z} sigma^{z}right)$$
from spin Orbital Coupling matrix in p-orbital basis, we know that the matrix elements between orbitals with different $l$ will vanish, e.g.
$$langle s|H_{SOC} |p_{x/y/z}rangle=0$$
since $L^{z/+/-}$ cannot change $l$. But in practice, some reference in research will gives SOC gap between orbitals with different $l$. I am confused of it.