The reduction of the neutron spin-orbit splitting $2p_{3/2} - 2p_{1/2}$ between the $^{41}$Ca and $^{35}$Si isotones is a unique feature throughout the chart of nuclides, as the spin-orbit splitting usually increases with $A$. Moreover, its way of decrease, gradual between $^{41}$Ca and $^{35}$Si or abrupt between $^{37}$S and $^{35}$Si, as well as its origin, caused by the weak binding energy of the $p$ states or by the sudden central proton density depletion in $^{35}$Si, are subject of debate. The results reported here using the self-consistent Covariant Energy Density Functional calculations with the DD-ME2 parametrization rather point to an abrupt, local decrease in $^{35}$Si, and to the large dominance of the central density depletion effect. It is concluded that weak binding, central density depletion as well as correlations must be taken into account to fully evaluate the amplitude and causes of this spin-orbit reduction.
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