  Consider three normalised cuspidal eigenforms of weight $2$ and prime level
$p$. Under the assumption that the global root number of the associated triple
product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of
the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the
modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component
of the Griffiths intermediate Jacobian. The same result holds with the complex
Abel-Jacobi map replaced by its \'etale counterpart. As an application, we
deduce torsion properties of Chow-Heegner points associated with modified
diagonal cycles on elliptic curves of prime level with split multiplicative
reduction. The approach also works in the case of composite square-free level.
