 Solution

At coincidence ${\varphi }_1={\varphi }_2$, the triplet state spatial factor vanishes, while the lowest singlet state has finite, uniform pair probability. Consequently a weak short-range repulsion raises the singlet state energy more strongly: antisymmetry suppresses close encounters in the triplet state. In the contact-interaction limit the triplet state contribution is exactly zero, whereas the singlet state shift is positive. A finite-range interaction gives a nonzero but suppressed triplet state shift. This explains a relative energy increase; it does not by itself establish a crossing or justify using unchanged noninteracting wavefunctions for arbitrarily strong repulsion.