Vanishing of the diagonal cycle for equal self-dual representations

Let $\pi$ be a self-dual automorphic representation, let $\pi_1=\pi_2=\pi_3=\pi$, and let $f$ range over $\boxtimes_{i=1}^3\pi_i^U$. Vanishing conjecture for equal self-dual representations. If

$$\epsilon(1/2,\pi)=-1,$$

then

$$[\Delta_3]_f=0$$

for every $f\in\boxtimes_{i=1}^3\pi_i^U$. Motivated by the generalized Gross–Kudla conjecture, this predicts vanishing of the modified diagonal cycle in the equal-representation case; the source presents it as an open conjecture.